#set document(title: "2.1 Use a General Strategy to Solve Linear Equations", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 2.1#h(0.6em)Use a General Strategy to Solve Linear Equations #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Before you get started, take this readiness quiz. Simplify: #math.equation(block: false, alt: "the fraction 3 over 2 open parenthesis 12 x plus 20 close parenthesis .")[$frac(3, 2) ( 12 x + 20 ) .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "18 x plus 30")[$18 x + 30$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "5 minus 2 open parenthesis n plus 1 close parenthesis .")[$5 − 2 ( n + 1 ) .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "3 minus 2 n")[$3 − 2 n$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the LCD of #math.equation(block: false, alt: "the fraction 5 over 6")[$frac(5, 6)$] and #math.equation(block: false, alt: "the fraction 1 over 4 .")[$frac(1, 4) .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "12")[$12$] ] ] === Solve Linear Equations Using a General Strategy Solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that makes it a true statement. Any value of the variable that makes the equation true is called a #strong[solution] to the equation. It is the answer to the puzzle! #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Solution of an Equation] A #strong[solution] of an equation is a value of a variable that makes a true statement when substituted into the equation. ] To determine whether a number is a solution to an equation, we substitute the value for the variable in the equation. If the resulting equation is a true statement, then the number is a solution of the equation. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Determine Whether a Number is a Solution to an Equation.] + Substitute the number for the variable in the equation. + Simplify the expressions on both sides of the equation. + Determine whether the resulting equation is true. - If it is true, the number is a solution. - If it is not true, the number is not a solution. ] #examplebox("Example 1")[][ Determine whether the values are solutions to the equation: #math.equation(block: false, alt: "5 y plus 3 equals 10 y minus 4 .")[$5 y + 3 = 10 y − 4 .$] ⓐ #math.equation(block: false, alt: "y equals the fraction 3 over 5")[$y = frac(3, 5)$] ⓑ #math.equation(block: false, alt: "y equals the fraction 7 over 5")[$y = frac(7, 5)$] #solutionbox[ Since a solution to an equation is a value of the variable that makes the equation true, begin by substituting the value of the solution for the variable. ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A linear algebraic equation is shown: 5y + 3 = 10y - 4, which involves the variable 'y' and constant terms on both sides of the equality sign, requiring simplification to solve for 'y'.], alt: "A linear algebraic equation is shown: 5y + 3 = 10y - 4, which involves the variable 'y' and constant terms on both sides of the equality sign, requiring simplification to solve for 'y'.", caption: none)]), [#figure(figph[The image shows the text 'Substitute 3/5 for y.' in a mathematical context, likely instructing the viewer to replace the variable 'y' with the fraction 3/5.], alt: "The image shows the text 'Substitute 3/5 for y.' in a mathematical context, likely instructing the viewer to replace the variable 'y' with the fraction 3/5.", caption: none)], [#figure(figph[The image displays the math problem 5(3/5) + 3 with a question mark over the equals sign, comparing it to 10(3/5) - 4.], alt: "The image displays the math problem 5(3/5) + 3 with a question mark over the equals sign, comparing it to 10(3/5) - 4.", caption: none)], [Multiply.], [#figure(figph[A mathematical equation asks whether 3 + 3 equals 6 - 4, with a question mark above the equal sign. Calculating both sides reveals that 3 + 3 = 6 and 6 - 4 = 2, so the equality is false.], alt: "A mathematical equation asks whether 3 + 3 equals 6 - 4, with a question mark above the equal sign. Calculating both sides reveals that 3 + 3 = 6 and 6 - 4 = 2, so the equality is false.", caption: none)], [Simplify.], [#figure(figph[The expression six is not equal to two.], alt: "The expression six is not equal to two.", caption: none)], )) Since #math.equation(block: false, alt: "y equals the fraction 3 over 5")[$y = frac(3, 5)$] does not result in a true equation, #math.equation(block: false, alt: "y equals the fraction 3 over 5")[$y = frac(3, 5)$] is not a solution to the equation #math.equation(block: false, alt: "5 y plus 3 equals 10 y minus 4 .")[$5 y + 3 = 10 y − 4 .$] ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical equation is displayed, reading '5y + 3 = 10y - 4' in black text on a white background, representing an algebraic problem.], alt: "A mathematical equation is displayed, reading '5y + 3 = 10y - 4' in black text on a white background, representing an algebraic problem.", caption: none)]), [#figure(figph[The text 'Substitute 7/5 for y.' is displayed, instructing to replace the variable 'y' with the fraction seven-fifths. The fraction is highlighted in red.], alt: "The text 'Substitute 7/5 for y.' is displayed, instructing to replace the variable 'y' with the fraction seven-fifths. The fraction is highlighted in red.", caption: none)], [#figure(figph[A math problem showing 5 times (7/5) + 3 being compared with 10 times (7/5) - 4, using an equals sign with a question mark.], alt: "A math problem showing 5 times (7/5) + 3 being compared with 10 times (7/5) - 4, using an equals sign with a question mark.", caption: none)], [Multiply.], [#figure(figph[A mathematical equation presented as 7 + 3 ?= 14 - 4, asking to verify if the two sides are equal.], alt: "A mathematical equation presented as 7 + 3 ?= 14 - 4, asking to verify if the two sides are equal.", caption: none)], [Simplify.], [#figure(figph[The number 10 is shown equal to 10, with a checkmark indicating correctness, against a white background.], alt: "The number 10 is shown equal to 10, with a checkmark indicating correctness, against a white background.", caption: none)], )) Since #math.equation(block: false, alt: "y equals the fraction 7 over 5")[$y = frac(7, 5)$] results in a true equation, #math.equation(block: false, alt: "y equals the fraction 7 over 5")[$y = frac(7, 5)$] is a solution to the equation #math.equation(block: false, alt: "5 y plus 3 equals 10 y minus 4 .")[$5 y + 3 = 10 y − 4 .$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Determine whether the values are solutions to the equation: #math.equation(block: false, alt: "9 y plus 2 equals 6 y plus 3 .")[$9 y + 2 = 6 y + 3 .$] ⓐ #math.equation(block: false, alt: "y equals the fraction 4 over 3")[$y = frac(4, 3)$] ⓑ #math.equation(block: false, alt: "y equals the fraction 1 over 3")[$y = frac(1, 3)$] #solutionbox[ ⓐ no ⓑ yes ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Determine whether the values are solutions to the equation: #math.equation(block: false, alt: "4 x minus 2 equals 2 x plus 1 .")[$4 x − 2 = 2 x + 1 .$] ⓐ #math.equation(block: false, alt: "x equals the fraction 3 over 2")[$x = frac(3, 2)$] ⓑ #math.equation(block: false, alt: "x equals minus the fraction 1 over 2")[$x = − frac(1, 2)$] #solutionbox[ ⓐ yes ⓑ no ] ] There are many types of equations that we will learn to solve. In this section we will focus on a #strong[linear equation]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Linear Equation] A #strong[linear equation] is an equation in one variable that can be written, where #emph[a] and #emph[b] are real numbers and #math.equation(block: false, alt: "a not equal to 0 ,")[$a ≠ 0 ,$] as: #math.equation(block: true, alt: "a x plus b equals 0")[$a x + b = 0$] ] To solve a linear equation it is a good idea to have an overall strategy that can be used to solve any linear equation. In the next example, we will give the steps of a general strategy for solving any linear equation. Simplifying each side of the equation as much as possible first makes the rest of the steps easier. #examplebox("Example 2")[How to Solve a Linear Equation Using a General Strategy][ Solve: #math.equation(block: false, alt: "7 open parenthesis n minus 3 close parenthesis minus 8 equals −15")[$7 ( n − 3 ) − 8 = −15$]. #solutionbox[ #figure(figph[Step 1 is to simplify each side of the equation, the product of 7 and the quantity n minus 3 minus 8 is equal to negative 15. Use the Distributive Property. The equation first simplifies to 7 n minus 21 minus 8 is equal to negative 15. Then it simplifies to 7 n minus 29 is equal to negative 15. Notice that each side of the equation is now simplified as much as possible.], alt: "Step 1 is to simplify each side of the equation, the product of 7 and the quantity n minus 3 minus 8 is equal to negative 15. Use the Distributive Property. The equation first simplifies to 7 n minus 21 minus 8 is equal to negative 15. Then it simplifies to 7 n minus 29 is equal to negative 15. Notice that each side of the equation is now simplified as much as possible.", caption: none) #figure(figph[Step 2 is to collect all variable terms on the left side of the equation, 7 n minus 29 is equal to negative 15. Notice there is nothing to do because all n’s are on the left side.], alt: "Step 2 is to collect all variable terms on the left side of the equation, 7 n minus 29 is equal to negative 15. Notice there is nothing to do because all n’s are on the left side.", caption: none) #figure(figph[Step 3 is to collect all constant terms on the other side of the equation, 7 n minus 29 is equal to negative 15. To get constants only on the right, add 29 to each side. The result is 7 n minus 29 plus 29 is equal to negative 15 plus 29. Simplify. The result is 7 n is equal to 14.], alt: "Step 3 is to collect all constant terms on the other side of the equation, 7 n minus 29 is equal to negative 15. To get constants only on the right, add 29 to each side. The result is 7 n minus 29 plus 29 is equal to negative 15 plus 29. Simplify. The result is 7 n is equal to 14.", caption: none) #figure(figph[Step 4 is to make the coefficient of the equation, 7 n is equal to 14, 1. Divide each side of the equation by 7. The result is 7 n divided by 7 is equal to 14 divided by 7. Simplify. The result is n is equal to 2.], alt: "Step 4 is to make the coefficient of the equation, 7 n is equal to 14, 1. Divide each side of the equation by 7. The result is 7 n divided by 7 is equal to 14 divided by 7. Simplify. The result is n is equal to 2.", caption: none) #figure(figph[Step 5 is to check the solution, n is equal to 2, by substituting into the equation, the product of 7 and the quantity n minus 3 minus 8 is equal to negative 15. Is the product of 7 and the quantity 2 minus 3 minus 8 equal to negative 15? Subtract. Is 7 times negative 1 minus 8 equal to negative 15? Is negative 7 minus 8 equal to negative 15. Negative 15 is equal to negative 15. The solution checks.], alt: "Step 5 is to check the solution, n is equal to 2, by substituting into the equation, the product of 7 and the quantity n minus 3 minus 8 is equal to negative 15. Is the product of 7 and the quantity 2 minus 3 minus 8 equal to negative 15? Subtract. Is 7 times negative 1 minus 8 equal to negative 15? Is negative 7 minus 8 equal to negative 15. Negative 15 is equal to negative 15. The solution checks.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "2 open parenthesis m minus 4 close parenthesis plus 3 equals −1 .")[$2 ( m − 4 ) + 3 = −1 .$] #solutionbox[ #math.equation(block: true, alt: "m equals 2")[$m = 2$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "5 open parenthesis a minus 3 close parenthesis plus 5 equals −10 .")[$5 ( a − 3 ) + 5 = −10 .$] #solutionbox[ #math.equation(block: true, alt: "a equals 0")[$a = 0$] ] ] These steps are summarized in the #strong[General Strategy for Solving Linear Equations] below. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Solve linear equations using a general strategy.] + Simplify each side of the equation as much as possible. #linebreak() Use the Distributive Property to remove any parentheses. #linebreak() Combine like terms. + Collect all the variable terms on one side of the equation. #linebreak() Use the Addition or Subtraction Property of Equality. + Collect all the constant terms on the other side of the equation. #linebreak() Use the Addition or Subtraction Property of Equality. + Make the coefficient of the variable term equal to 1. #linebreak() Use the Multiplication or Division Property of Equality. #linebreak() State the solution to the equation. + Check the solution. #linebreak() Substitute the solution into the original equation to make sure the result is a true statement. ] #examplebox("Example 3")[][ Solve: #math.equation(block: false, alt: "the fraction 2 over 3 open parenthesis 3 m minus 6 close parenthesis equals 5 minus m .")[$frac(2, 3) ( 3 m − 6 ) = 5 − m .$] #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, [], [#figure(figph[A mathematical equation is shown, displaying (2/3)(3m - 6) = 5 - m, with 'm' representing a variable. The expression on the left side involves a fraction multiplied by a binomial, while the right side shows a constant minus the variable.], alt: "A mathematical equation is shown, displaying (2/3)(3m - 6) = 5 - m, with 'm' representing a variable. The expression on the left side involves a fraction multiplied by a binomial, while the right side shows a constant minus the variable.", caption: none)], [], [Distribute.], [#figure(figph[A mathematical equation displays '2m - 4 = 5 - m' in a bold, dark gray font against a plain white background.], alt: "A mathematical equation displays '2m - 4 = 5 - m' in a bold, dark gray font against a plain white background.", caption: none)], [], [Add #emph[m] to both sides to get the variables only on the left.], [#figure(figph[An algebraic equation is shown: 2m + m - 4 = 5 - m + m. Variables and constants are displayed in a clean, sans-serif font, with some 'm' terms highlighted in red.], alt: "An algebraic equation is shown: 2m + m - 4 = 5 - m + m. Variables and constants are displayed in a clean, sans-serif font, with some 'm' terms highlighted in red.", caption: none)], [], [Simplify.], [#figure(figph[A mathematical equation is displayed on a white background. The equation reads '3m - 4 = 5' in black text, solving for the variable 'm'.], alt: "A mathematical equation is displayed on a white background. The equation reads '3m - 4 = 5' in black text, solving for the variable 'm'.", caption: none)], [], [Add 4 to both sides to get constants only on the right.], [#figure(figph[An algebraic equation showing 3m - 4 + 4 = 5 + 4, illustrating the addition of 4 to both sides to solve for 'm'.], alt: "An algebraic equation showing 3m - 4 + 4 = 5 + 4, illustrating the addition of 4 to both sides to solve for 'm'.", caption: none)], [], [Simplify.], [#figure(figph[A simple linear equation '3m = 9' is displayed, demonstrating a basic algebraic problem.], alt: "A simple linear equation '3m = 9' is displayed, demonstrating a basic algebraic problem.", caption: none)], [], [Divide both sides by three.], [#figure(figph[A mathematical equation shows both sides being divided by 3 to solve for 'm'. The equation is 3m/3 = 9/3, with the denominator '3' on both sides highlighted in red.], alt: "A mathematical equation shows both sides being divided by 3 to solve for 'm'. The equation is 3m/3 = 9/3, with the denominator '3' on both sides highlighted in red.", caption: none)], [], [Simplify.], [#figure(figph[The image displays a simple mathematical equation, 'm = 3', rendered in black text on a plain white background. The equation indicates that the variable 'm' is equal to the number '3'.], alt: "The image displays a simple mathematical equation, 'm = 3', rendered in black text on a plain white background. The equation indicates that the variable 'm' is equal to the number '3'.", caption: none)], [], [Check:], [#figure(figph[A mathematical equation is displayed: 2/3 multiplied by the quantity 3m minus 6, which is equal to 5 minus m. The equation is (2/3)(3m - 6) = 5 - m.], alt: "A mathematical equation is displayed: 2/3 multiplied by the quantity 3m minus 6, which is equal to 5 minus m. The equation is (2/3)(3m - 6) = 5 - m.", caption: none)], [], [Let #math.equation(block: false, alt: "m equals 3.")[$m = 3.$]], [#figure(figph[A mathematical equation shown as (2/3)(3 \* 3 - 6) =? 5 - 3. The numbers '3' (in '3 \* 3') and '3' (in '5 - 3') are highlighted in red, indicating a verification or problem-solving context.], alt: "A mathematical equation shown as (2/3)(3 * 3 - 6) =? 5 - 3. The numbers '3' (in '3 * 3') and '3' (in '5 - 3') are highlighted in red, indicating a verification or problem-solving context.", caption: none)], [], [], [#figure(figph[A mathematical equation questions if (2/3)(9-6) is equal to 2.], alt: "A mathematical equation questions if (2/3)(9-6) is equal to 2.", caption: none)], [], [], [#figure(figph[A mathematical problem asking whether two-thirds multiplied by three equals two. The expression is (2/3)(3) ?= 2, with the question mark indicating an inquiry into the equality.], alt: "A mathematical problem asking whether two-thirds multiplied by three equals two. The expression is (2/3)(3) ?= 2, with the question mark indicating an inquiry into the equality.", caption: none)], [], [], [#figure(figph[The number two equals two, confirmed with a checkmark, representing a correct and verified mathematical statement or a simple affirmation of equality.], alt: "The number two equals two, confirmed with a checkmark, representing a correct and verified mathematical statement or a simple affirmation of equality.", caption: none)], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction 1 over 3 open parenthesis 6 u plus 3 close parenthesis equals 7 minus u .")[$frac(1, 3) ( 6 u + 3 ) = 7 − u .$] #solutionbox[ #math.equation(block: true, alt: "u equals 2")[$u = 2$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction 2 over 3 open parenthesis 9 x minus 12 close parenthesis equals 8 plus 2 x .")[$frac(2, 3) ( 9 x − 12 ) = 8 + 2 x .$] #solutionbox[ #math.equation(block: true, alt: "x equals 4")[$x = 4$] ] ] We can solve equations by getting all the #strong[variable] terms to either side of the #strong[equal sign]. By collecting the variable terms on the side where the #strong[coefficient] of the variable is larger, we avoid working with some negatives. This will be a good strategy when we solve inequalities later in this chapter. It also helps us prevent errors with negatives. #examplebox("Example 4")[][ Solve: #math.equation(block: false, alt: "4 open parenthesis x minus 1 close parenthesis minus 2 equals 5 open parenthesis 2 x plus 3 close parenthesis plus 6 .")[$4 ( x − 1 ) − 2 = 5 ( 2 x + 3 ) + 6 .$] #solutionbox[ #figure(table( columns: 5, align: left, inset: 6pt, [], [], [], [#figure(figph[A mathematical equation is displayed: 4(x - 1) - 2 = 5(2x + 3) + 6. This is a linear equation with one variable, x, and involves distribution and simplification to solve for x.], alt: "A mathematical equation is displayed: 4(x - 1) - 2 = 5(2x + 3) + 6. This is a linear equation with one variable, x, and involves distribution and simplification to solve for x.", caption: none)], [], [Distribute.], [], [], [#figure(figph[A mathematical equation is displayed: 4x - 4 - 2 = 10x + 15 + 6. It is a linear equation with variables on both sides, and it involves subtraction and addition operations.], alt: "A mathematical equation is displayed: 4x - 4 - 2 = 10x + 15 + 6. It is a linear equation with variables on both sides, and it involves subtraction and addition operations.", caption: none)], [], [Combine like terms.], [], [], [#figure(figph[The image displays an algebraic equation: 4x - 6 = 10x + 21.], alt: "The image displays an algebraic equation: 4x - 6 = 10x + 21.", caption: none)], [], [Subtract #math.equation(block: false, alt: "4 x")[$4 x$] from each side to get the variables only on #linebreak() the right since #math.equation(block: false, alt: "10 greater than 4 .")[$10 > 4 .$]], [], [], [#figure(figph[An algebra equation: 4x - 4x - 6 = 10x - 4x + 21. The -4x terms on both sides are highlighted in red, indicating they can be canceled or combined to simplify the equation.], alt: "An algebra equation: 4x - 4x - 6 = 10x - 4x + 21. The -4x terms on both sides are highlighted in red, indicating they can be canceled or combined to simplify the equation.", caption: none)], [], [Simplify.], [], [], [#figure(figph[A mathematical equation is displayed, showing -6 = 6x + 21, indicating an algebraic problem to solve for x.], alt: "A mathematical equation is displayed, showing -6 = 6x + 21, indicating an algebraic problem to solve for x.", caption: none)], [], [Subtract 21 from each side to get the constants on left.], [], [], [#figure(figph[A mathematical equation, -6 - 21 = 6x + 21 - 21, showing the step of subtracting 21 from both sides (highlighted in red) to isolate the variable term.], alt: "A mathematical equation, -6 - 21 = 6x + 21 - 21, showing the step of subtracting 21 from both sides (highlighted in red) to isolate the variable term.", caption: none)], [], [Simplify.], [], [], [#figure(figph[An algebraic equation showing -27 equals 6 times x.], alt: "An algebraic equation showing -27 equals 6 times x.", caption: none)], [], [Divide both sides by 6.], [], [], [#figure(figph[An equation displaying -27/6 = 6x/6, where the common denominator 6 is highlighted in red.], alt: "An equation displaying -27/6 = 6x/6, where the common denominator 6 is highlighted in red.", caption: none)], [], [Simplify.], [], [], [#figure(figph[A mathematical equation shows a fraction negative nine over two equals x, written as -9/2 = x.], alt: "A mathematical equation shows a fraction negative nine over two equals x, written as -9/2 = x.", caption: none)], [], [Check:], [#figure(figph[A mathematical equation is displayed against a white background: 4(x - 1) - 2 = 5(2x + 3) + 6.], alt: "A mathematical equation is displayed against a white background: 4(x - 1) - 2 = 5(2x + 3) + 6.", caption: none)], [], [], [], [Let #math.equation(block: false, alt: "x equals minus the fraction 9 over 2 .")[$x = − frac(9, 2) .$]], [#figure(figph[A mathematical equation compares two expressions using a question mark over an equals sign: 4(-9/2 - 1) - 2 and 5(2(-9/2) + 3) + 6. The term -9/2 is highlighted in red.], alt: "A mathematical equation compares two expressions using a question mark over an equals sign: 4(-9/2 - 1) - 2 and 5(2(-9/2) + 3) + 6. The term -9/2 is highlighted in red.", caption: none)], [], [], [], [], [#figure(figph[A mathematical equation is shown: 4(-11/2) - 2 =? 5(-9 + 3) + 6, asking to verify if the left side equals the right side.], alt: "A mathematical equation is shown: 4(-11/2) - 2 =? 5(-9 + 3) + 6, asking to verify if the left side equals the right side.", caption: none)], [], [], [], [], [#figure(figph[A mathematical equation on a white background asks to verify if -22 - 2 equals 5 multiplied by -6, plus 6.], alt: "A mathematical equation on a white background asks to verify if -22 - 2 equals 5 multiplied by -6, plus 6.", caption: none)], [], [], [], [], [#figure(figph[The image presents a numerical equation: -24 ?= -30 + 6, questioning the equality between the two sides.], alt: "The image presents a numerical equation: -24 ?= -30 + 6, questioning the equality between the two sides.", caption: none)], [], [], [], [], [#figure(figph[A mathematical equation shows -24 equals -24, followed by a checkmark, indicating the equality is correct.], alt: "A mathematical equation shows -24 equals -24, followed by a checkmark, indicating the equality is correct.", caption: none)], [], [], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "6 open parenthesis p minus 3 close parenthesis minus 7 equals 5 open parenthesis 4 p plus 3 close parenthesis minus 12 .")[$6 ( p − 3 ) − 7 = 5 ( 4 p + 3 ) − 12 .$] #solutionbox[ #math.equation(block: true, alt: "p equals −2")[$p = −2$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "8 open parenthesis q plus 1 close parenthesis minus 5 equals 3 open parenthesis 2 q minus 4 close parenthesis minus 1 .")[$8 ( q + 1 ) − 5 = 3 ( 2 q − 4 ) − 1 .$] #solutionbox[ #math.equation(block: true, alt: "q equals −8")[$q = −8$] ] ] #examplebox("Example 5")[][ Solve: #math.equation(block: false, alt: "10 [ 3 minus 8 open parenthesis 2 s minus 5 close parenthesis ] equals 15 open parenthesis 40 minus 5 s close parenthesis .")[$10 [ 3 − 8 ( 2 s − 5 ) ] = 15 ( 40 − 5 s ) .$] #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, [], [#figure(figph[A mathematical equation shows 10 multiplied by the expression \[3 minus 8 times (2s minus 5)\], which equals 15 multiplied by the expression (40 minus 5s).], alt: "A mathematical equation shows 10 multiplied by the expression [3 minus 8 times (2s minus 5)], which equals 15 multiplied by the expression (40 minus 5s).", caption: none)], [], [Simplify from the innermost parentheses first.], [#figure(figph[An algebraic equation is presented: 10\[3 - 16s + 40\] = 15(40 - 5s).], alt: "An algebraic equation is presented: 10[3 - 16s + 40] = 15(40 - 5s).", caption: none)], [], [Combine like terms in the brackets.], [#figure(figph[A mathematical equation is displayed: 10\[43 - 16s\] = 15(40 - 5s). The equation features numbers, variables, and arithmetic operations within brackets and parentheses, set against a white background.], alt: "A mathematical equation is displayed: 10[43 - 16s] = 15(40 - 5s). The equation features numbers, variables, and arithmetic operations within brackets and parentheses, set against a white background.", caption: none)], [], [Distribute.], [#figure(figph[A mathematical equation is displayed on a white background: 430 - 160s = 600 - 75s. The equation involves numbers and the variable 's' on both sides of the equality sign.], alt: "A mathematical equation is displayed on a white background: 430 - 160s = 600 - 75s. The equation involves numbers and the variable 's' on both sides of the equality sign.", caption: none)], [], [Add #math.equation(block: false, alt: "160 s")[$160 s$] to both sides to get the #linebreak() variables to the right.], [#figure(figph[An algebraic equation showing a step in solving for 's': 430 - 160s + 160s = 600 - 75s + 160s. The term '+ 160s' is highlighted in red on both sides of the equality.], alt: "An algebraic equation showing a step in solving for 's': 430 - 160s + 160s = 600 - 75s + 160s. The term '+ 160s' is highlighted in red on both sides of the equality.", caption: none)], [], [Simplify.], [#figure(figph[The mathematical equation 430 = 600 + 85s is displayed.], alt: "The mathematical equation 430 = 600 + 85s is displayed.", caption: none)], [], [Subtract 600 from both sides to get the #linebreak() constants to the left.], [#figure(figph[A mathematical equation, 430 - 600 = 600 + 85s - 600, with the 600 terms highlighted in red, suggesting an operation like cancellation or subtraction.], alt: "A mathematical equation, 430 - 600 = 600 + 85s - 600, with the 600 terms highlighted in red, suggesting an operation like cancellation or subtraction.", caption: none)], [], [Simplify.], [#figure(figph[A mathematical equation is displayed, showing '-170 = 85s' against a white background.], alt: "A mathematical equation is displayed, showing '-170 = 85s' against a white background.", caption: none)], [], [Divide both sides by 85.], [#figure(figph[A mathematical equation shows both sides being divided by 85 to solve for 's', with -170/85 on the left and 85s/85 on the right, setting up for -2 = s.], alt: "A mathematical equation shows both sides being divided by 85 to solve for 's', with -170/85 on the left and 85s/85 on the right, setting up for -2 = s.", caption: none)], [], [Simplify.], [#figure(figph[A minimalistic image displaying a mathematical equation: -2 = S.], alt: "A minimalistic image displaying a mathematical equation: -2 = S.", caption: none)], [], [Check:], [#figure(figph[A mathematical equation: 10\[3 - 8(2s - 5)\] = 15(40 - 5s), presented in a clear, dark font against a white background.], alt: "A mathematical equation: 10[3 - 8(2s - 5)] = 15(40 - 5s), presented in a clear, dark font against a white background.", caption: none)], [], [Let #math.equation(block: false, alt: "s equals minus 2.")[$s = − 2.$]], [#figure(figph[A mathematical expression featuring nested operations, including multiplication, subtraction, and a power, with some numbers in red indicating negative values, separated by a question mark and followed by another expression.], alt: "A mathematical expression featuring nested operations, including multiplication, subtraction, and a power, with some numbers in red indicating negative values, separated by a question mark and followed by another expression.", caption: none)], [], [], [#figure(figph[A mathematical expression reads 10 multiplied by the square of \[3 minus 8 times (-4 minus 5)\], which is followed by an inequality symbol indicating 'greater than or equal to', then 15 multiplied by (40 plus 10).], alt: "A mathematical expression reads 10 multiplied by the square of [3 minus 8 times (-4 minus 5)], which is followed by an inequality symbol indicating 'greater than or equal to', then 15 multiplied by (40 plus 10).", caption: none)], [], [], [#figure(figph[A mathematical inequality expression displays '10\[3 - 8(-9)\]²' on the left side, which is greater than or equal to '15(50)' on the right side. The equation shows a combination of numbers, operators, brackets, and an exponent.], alt: "A mathematical inequality expression displays '10[3 - 8(-9)]²' on the left side, which is greater than or equal to '15(50)' on the right side. The equation shows a combination of numbers, operators, brackets, and an exponent.", caption: none)], [], [], [#figure(figph[A mathematical equation asks whether 10 multiplied by the square of (3 plus 72) is equal to 750.], alt: "A mathematical equation asks whether 10 multiplied by the square of (3 plus 72) is equal to 750.", caption: none)], [], [], [#figure(figph[A mathematical expression 10(75)^2 with a question mark over the equal sign, asking if it equals 750, is displayed against a white background.], alt: "A mathematical expression 10(75)^2 with a question mark over the equal sign, asking if it equals 750, is displayed against a white background.", caption: none)], [], [], [#figure(figph[A simple equation 750 = 750 is displayed with a checkmark, confirming its correctness. This image humorously represents agreement, a solved problem, or an undeniable truth.], alt: "A simple equation 750 = 750 is displayed with a checkmark, confirming its correctness. This image humorously represents agreement, a solved problem, or an undeniable truth.", caption: none)], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "6 [ 4 minus 2 open parenthesis 7 y minus 1 close parenthesis ] equals 8 open parenthesis 13 minus 8 y close parenthesis .")[$6 [ 4 − 2 ( 7 y − 1 ) ] = 8 ( 13 − 8 y ) .$] #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 17 over 5")[$y = − frac(17, 5)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "12 [ 1 minus 5 open parenthesis 4 z minus 1 close parenthesis ] equals 3 open parenthesis 24 plus 11 z close parenthesis .")[$12 [ 1 − 5 ( 4 z − 1 ) ] = 3 ( 24 + 11 z ) .$] #solutionbox[ #math.equation(block: true, alt: "z equals 0")[$z = 0$] ] ] === Classify Equations Whether or not an equation is true depends on the value of the variable. The equation #math.equation(block: false, alt: "7 x plus 8 equals −13")[$7 x + 8 = −13$] is true when we replace the variable, #emph[x], with the value #math.equation(block: false, alt: "−3 ,")[$−3 ,$] but not true when we replace #emph[x] with any other value. An equation like this is called a #strong[conditional equation]. All the equations we have solved so far are conditional equations. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Conditional Equation] An equation that is true for one or more values of the variable and false for all other values of the variable is a #strong[conditional equation]. ] Now let’s consider the equation #math.equation(block: false, alt: "7 y plus 14 equals 7 open parenthesis y plus 2 close parenthesis .")[$7 y + 14 = 7 ( y + 2 ) .$] Do you recognize that the left side and the right side are equivalent? Let’s see what happens when we solve for #emph[y]. Solve: #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The image displays the mathematical equation 7y + 14 = 7(y + 2), which illustrates the distributive property or factoring a common term.], alt: "The image displays the mathematical equation 7y + 14 = 7(y + 2), which illustrates the distributive property or factoring a common term.", caption: none)]), [Distribute.], [#figure(figph[A mathematical equation displays '7y + 14 = 7y + 14' in a clear, sans-serif font against a white background, representing an algebraic identity.], alt: "A mathematical equation displays '7y + 14 = 7y + 14' in a clear, sans-serif font against a white background, representing an algebraic identity.", caption: none)], [Subtract #math.equation(block: false, alt: "7 y")[$7 y$] to each side to get the #math.equation(block: false, alt: "y ’ s")[$y ’ "s"$] to one side.], [#figure(figph[The equation 7y - 7y + 14 = 7y - 7y + 14, which simplifies to 14 = 14, is a mathematical identity.], alt: "The equation 7y - 7y + 14 = 7y - 7y + 14, which simplifies to 14 = 14, is a mathematical identity.", caption: none)], [Simplify—the #emph[y]’s are eliminated.], [#figure(figph[Text displaying the simple mathematical equality '14 = 14' on a clear white background.], alt: "Text displaying the simple mathematical equality '14 = 14' on a clear white background.", caption: none)], [], [But #math.equation(block: false, alt: "14 equals 14")[$14 = 14$] is true.], )) This means that the equation #math.equation(block: false, alt: "7 y plus 14 equals 7 open parenthesis y plus 2 close parenthesis")[$7 y + 14 = 7 ( y + 2 )$] is true for any value of #emph[y]. We say the solution to the equation is all of the real numbers. An equation that is true for any value of the variable is called an #strong[identity]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Identity] An equation that is true for any value of the variable is called an #strong[identity]. The solution of an identity is all real numbers. ] What happens when we solve the equation #math.equation(block: false, alt: "−8 z equals −8 z plus 9 ?")[$−8 z = −8 z + 9 ?$] Solve: #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The image displays the algebraic equation -8z = -8z + 9, which simplifies to 0 = 9, indicating that the equation has no solution.], alt: "The image displays the algebraic equation -8z = -8z + 9, which simplifies to 0 = 9, indicating that the equation has no solution.", caption: none)]), [Add #math.equation(block: false, alt: "8 z")[$8 z$] to both sides to leave the constant alone on the right.], [#figure(figph[The equation -8z + 8z = -8z + 8z + 9 is shown. This simplifies to 0 = 9, indicating that this equation has no solution.], alt: "The equation -8z + 8z = -8z + 8z + 9 is shown. This simplifies to 0 = 9, indicating that this equation has no solution.", caption: none)], [Simplify—the #math.equation(block: false, alt: "z ’ s")[$z ’ "s"$] are eliminated.], [#figure(figph[The mathematical expression '0 is not equal to 9' is displayed in black text on a white background.], alt: "The mathematical expression '0 is not equal to 9' is displayed in black text on a white background.", caption: none)], [], [But #math.equation(block: false, alt: "0 not equal to 9 .")[$0 ≠ 9 .$]], )) Solving the equation #math.equation(block: false, alt: "−8 z equals −8 z plus 9")[$−8 z = −8 z + 9$] led to the false statement #math.equation(block: false, alt: "0 equals 9 .")[$0 = 9 .$] The equation #math.equation(block: false, alt: "−8 z equals −8 z plus 9")[$−8 z = −8 z + 9$] will not be true for any value of #emph[z]. It has no solution. An equation that has no solution, or that is false for all values of the variable, is called a #strong[contradiction]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Contradiction] An equation that is false for all values of the variable is called a #strong[contradiction]. A contradiction has no solution. ] The next few examples will ask us to classify an equation as conditional, an identity, or as a contradiction. #examplebox("Example 6")[][ Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: #math.equation(block: false, alt: "6 open parenthesis 2 n minus 1 close parenthesis plus 3 equals 2 n minus 8 plus 5 open parenthesis 2 n plus 1 close parenthesis .")[$6 ( 2 n − 1 ) + 3 = 2 n − 8 + 5 ( 2 n + 1 ) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical equation is displayed: 6(2n - 1) + 3 = 2n - 8 + 5(2n + 1). The equation involves linear expressions with the variable 'n' on both sides of the equality.], alt: "A mathematical equation is displayed: 6(2n - 1) + 3 = 2n - 8 + 5(2n + 1). The equation involves linear expressions with the variable 'n' on both sides of the equality.", caption: none)]), [Distribute.], [#figure(figph[An algebraic equation is shown, displaying '12n - 6 + 3 = 2n - 8 + 10n + 5'.], alt: "An algebraic equation is shown, displaying '12n - 6 + 3 = 2n - 8 + 10n + 5'.", caption: none)], [Combine like terms.], [#figure(figph[A mathematical equation is displayed, showing '12n - 3 = 12n - 3' in a simple, clear font on a white background. The equation represents an identity, as both sides are identical.], alt: "A mathematical equation is displayed, showing '12n - 3 = 12n - 3' in a simple, clear font on a white background. The equation represents an identity, as both sides are identical.", caption: none)], [Subtract #math.equation(block: false, alt: "12 n")[$12 n$] from each side to get the #emph[n]’s to one side.], [#figure(figph[An algebraic identity: 12n - 12n - 3 = 12n - 12n - 3 simplifies to -3 = -3, true for any value of 'n'.], alt: "An algebraic identity: 12n - 12n - 3 = 12n - 12n - 3 simplifies to -3 = -3, true for any value of 'n'.", caption: none)], [Simplify.], [#figure(figph[A mathematical equation showing -3 equals -3 is displayed on a white background.], alt: "A mathematical equation showing -3 equals -3 is displayed on a white background.", caption: none)], [This is a true statement.], [The equation is an identity.], [], [The solution is all real numbers.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: #math.equation(block: false, alt: "4 plus 9 open parenthesis 3 x minus 7 close parenthesis equals −42 x minus 13 plus 23 open parenthesis 3 x minus 2 close parenthesis .")[$4 + 9 ( 3 x − 7 ) = −42 x − 13 + 23 ( 3 x − 2 ) .$] #solutionbox[ identity; all real numbers ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: #math.equation(block: false, alt: "8 open parenthesis 1 minus 3 x close parenthesis plus 15 open parenthesis 2 x plus 7 close parenthesis equals 2 open parenthesis x plus 50 close parenthesis plus 4 open parenthesis x plus 3 close parenthesis plus 1 .")[$8 ( 1 − 3 x ) + 15 ( 2 x + 7 ) = 2 ( x + 50 ) + 4 ( x + 3 ) + 1 .$] #solutionbox[ identity; all real numbers ] ] #examplebox("Example 7")[][ Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: #math.equation(block: false, alt: "8 plus 3 open parenthesis a minus 4 close parenthesis equals 0 .")[$8 + 3 ( a − 4 ) = 0 .$] #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, [], [], [#figure(figph[A mathematical equation is displayed: 8 + 3(a - 4) = 0. The equation involves numbers, variables, and basic arithmetic operations set equal to zero.], alt: "A mathematical equation is displayed: 8 + 3(a - 4) = 0. The equation involves numbers, variables, and basic arithmetic operations set equal to zero.", caption: none)], [Distribute.], [], [#figure(figph[A mathematical equation reads 8 + 3a - 12 = 0 on a white background, in a close-up shot.], alt: "A mathematical equation reads 8 + 3a - 12 = 0 on a white background, in a close-up shot.", caption: none)], [Combine like terms.], [], [#figure(figph[A mathematical equation displays '3q - 4 = 0' on a white background.], alt: "A mathematical equation displays '3q - 4 = 0' on a white background.", caption: none)], [Add 4 to both sides.], [], [#figure(figph[A mathematical equation reads '3a - 4 + 4 = 0 + 4', demonstrating the addition property of equality where 4 is added to both sides of the equation. The numbers 4 and the plus signs are highlighted in red.], alt: "A mathematical equation reads '3a - 4 + 4 = 0 + 4', demonstrating the addition property of equality where 4 is added to both sides of the equation. The numbers 4 and the plus signs are highlighted in red.", caption: none)], [Simplify.], [], [#figure(figph[The mathematical equation '3a = 4' is displayed on a white background.], alt: "The mathematical equation '3a = 4' is displayed on a white background.", caption: none)], [Divide.], [], [#figure(figph[A mathematical equation shows '3a over 3 equals 4 over 3', with the number 3 in the denominator of both fractions highlighted in red, indicating a division operation.], alt: "A mathematical equation shows '3a over 3 equals 4 over 3', with the number 3 in the denominator of both fractions highlighted in red, indicating a division operation.", caption: none)], [Simplify.], [], [#figure(figph[The mathematical equation displays 'a = 4/3' in a simple, clear font on a white background, representing the value of the variable 'a' as a fraction.], alt: "The mathematical equation displays 'a = 4/3' in a simple, clear font on a white background, representing the value of the variable 'a' as a fraction.", caption: none)], [The equation is true when #math.equation(block: false, alt: "a equals the fraction 4 over 3 .")[$a = frac(4, 3) . #h(2em)$]], [This is a conditional equation.], [], [], [The solution is #math.equation(block: false, alt: "a equals the fraction 4 over 3 .")[$a = frac(4, 3) .$]], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: #math.equation(block: false, alt: "11 open parenthesis q plus 3 close parenthesis minus 5 equals 19 .")[$11 ( q + 3 ) − 5 = 19 .$] #solutionbox[ conditional equation; #math.equation(block: false, alt: "q equals minus the fraction 9 over 11")[$q = − frac(9, 11)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: #math.equation(block: false, alt: "6+14 open parenthesis k minus 8 close parenthesis equals 95 .")[$6+14 ( k − 8 ) = 95 .$] #solutionbox[ conditional equation; #math.equation(block: false, alt: "k equals the fraction 201 over 14")[$k = frac(201, 14)$] ] ] #examplebox("Example 8")[][ Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: #math.equation(block: false, alt: "5 m plus 3 open parenthesis 9 plus 3 m close parenthesis equals 2 open parenthesis 7 m minus 11 close parenthesis .")[$5 m + 3 ( 9 + 3 m ) = 2 ( 7 m − 11 ) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[An algebraic equation is shown: 5m + 3(9 + 3m) = 2(7m - 11).], alt: "An algebraic equation is shown: 5m + 3(9 + 3m) = 2(7m - 11).", caption: none)]), [Distribute.], [#figure(figph[A mathematical equation is displayed, showing '5m + 27 + 9m = 14m - 22' in white text on a black background, representing a linear equation with variable 'm' to be solved.], alt: "A mathematical equation is displayed, showing '5m + 27 + 9m = 14m - 22' in white text on a black background, representing a linear equation with variable 'm' to be solved.", caption: none)], [Combine like terms.], [#figure(figph[A mathematical equation is displayed, showing '14m + 27 = 14m - 22' in black text against a white background.], alt: "A mathematical equation is displayed, showing '14m + 27 = 14m - 22' in black text against a white background.", caption: none)], [Subtract #math.equation(block: false, alt: "14 m")[$14 m$] from both sides.], [#figure(figph[An algebraic equation: 14m + 27 - 14m = 14m - 22 - 14m, illustrating the step of subtracting 14m from both sides, with the subtracted terms highlighted in red.], alt: "An algebraic equation: 14m + 27 - 14m = 14m - 22 - 14m, illustrating the step of subtracting 14m from both sides, with the subtracted terms highlighted in red.", caption: none)], [Simplify.], [#figure(figph[A mathematical expression '27 ≠ -22' is displayed against a white background, indicating that the number 27 is not equal to -22.], alt: "A mathematical expression '27 ≠ -22' is displayed against a white background, indicating that the number 27 is not equal to -22.", caption: none)], [But #math.equation(block: false, alt: "27 not equal to − 22 .")[$27 ≠ "−" 22 .$]], [The equation is a contradiction.], [], [It has no solution.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: #math.equation(block: false, alt: "12 c plus 5 open parenthesis 5 plus 3 c close parenthesis equals 3 open parenthesis 9 c minus 4 close parenthesis .")[$12 c + 5 ( 5 + 3 c ) = 3 ( 9 c − 4 ) .$] #solutionbox[ contradiction; no solution ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Classify the equation as a conditional equation, an identity, or a contradiction and then state the solution: #math.equation(block: false, alt: "4 open parenthesis 7 d plus 18 close parenthesis equals 13 open parenthesis 3 d minus 2 close parenthesis minus 11 d .")[$4 ( 7 d + 18 ) = 13 ( 3 d − 2 ) − 11 d .$] #solutionbox[ contradiction; no solution ] ] We summarize the methods for classifying equations in the table. #figure(table( columns: 3, align: left, inset: 6pt, table.header([Type of equation], [What happens when you solve it?], [Solution]), [#strong[Conditional Equation]], [True for one or more values of the variables and false for all other values], [One or more values], [#strong[Identity]], [True for any value of the variable], [All real numbers], [#strong[Contradiction]], [False for all values of the variable], [No solution], )) === Solve Equations with Fraction or Decimal Coefficients We could use the General Strategy to solve the next example. This method would work fine, but many students do not feel very confident when they see all those fractions. So, we are going to show an alternate method to solve equations with fractions. This alternate method eliminates the fractions. We will apply the Multiplication Property of Equality and multiply both sides of an equation by the #strong[least common denominator] (LCD) of all the fractions in the equation. The result of this operation will be a new equation, equivalent to the first, but without fractions. This process is called #emph[clearing] the equation of fractions. To clear an equation of decimals, we think of all the decimals in their fraction form and then find the LCD of those denominators. #examplebox("Example 9")[How to Solve Equations with Fraction or Decimal Coefficients][ Solve: #math.equation(block: false, alt: "the fraction 1 over 12 x plus the fraction 5 over 6 equals the fraction 3 over 4 .")[$frac(1, 12) x + frac(5, 6) = frac(3, 4) .$] #solutionbox[ #figure(figph[Step 1 is to find the least common denominator of all the fractions and decimals in the equation, one-twelfth x plus five-sixth is equal to three-fourths. What is the L C D of one-twelfth, five-sixths, and three-fourths? The L C D is equal to 12.], alt: "Step 1 is to find the least common denominator of all the fractions and decimals in the equation, one-twelfth x plus five-sixth is equal to three-fourths. What is the L C D of one-twelfth, five-sixths, and three-fourths? The L C D is equal to 12.", caption: none) #figure(figph[Step 2 is multiply both sides of the equation by the L C D. This clears the fractions and decimals. Multiply both sides of the equation by the L C D, 12. The result is 12 times the quantity one-twelfth x plus five-sixths is equal to 12 times three-fourths. Use the Distributive Property. The result is 12 times one-twelfth x plus 12 times five-sixths is equal to 12 times three-fourths. Simplify. The result is x plus 10 is equal to 9. Notice there are no more fractions.], alt: "Step 2 is multiply both sides of the equation by the L C D. This clears the fractions and decimals. Multiply both sides of the equation by the L C D, 12. The result is 12 times the quantity one-twelfth x plus five-sixths is equal to 12 times three-fourths. Use the Distributive Property. The result is 12 times one-twelfth x plus 12 times five-sixths is equal to 12 times three-fourths. Simplify. The result is x plus 10 is equal to 9. Notice there are no more fractions.", caption: none) #figure(figph[Step 3 is to solve using the General Strategy for Solving Linear Equations. To isolate the variable term, subtract 10. The result is x plus 10 minus 10 is equal to 9 minus 10. Simplify. The result is x is equal to negative 1. Check the solution. Substitute negative into the original equation one-twelfth x plus five-sixths is equal to three-fourth. Is one-twelfth times negative 1 plus five-sixths equal to three-fourths? Is negative one-twelfth plus five-sixths equal to three-fourths? Is negative one-twelfth plus ten-twelfths equal to nine-twelfths? Is nine-twelfths equal to nine-twelfths? Yes. The solution checks.], alt: "Step 3 is to solve using the General Strategy for Solving Linear Equations. To isolate the variable term, subtract 10. The result is x plus 10 minus 10 is equal to 9 minus 10. Simplify. The result is x is equal to negative 1. Check the solution. Substitute negative into the original equation one-twelfth x plus five-sixths is equal to three-fourth. Is one-twelfth times negative 1 plus five-sixths equal to three-fourths? Is negative one-twelfth plus five-sixths equal to three-fourths? Is negative one-twelfth plus ten-twelfths equal to nine-twelfths? Is nine-twelfths equal to nine-twelfths? Yes. The solution checks.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction 1 over 4 x plus the fraction 1 over 2 equals the fraction 5 over 8 .")[$frac(1, 4) x + frac(1, 2) = frac(5, 8) .$] #solutionbox[ #math.equation(block: true, alt: "x equals the fraction 1 over 2")[$x = frac(1, 2)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction 1 over 8 x plus the fraction 1 over 2 equals the fraction 1 over 4 .")[$frac(1, 8) x + frac(1, 2) = frac(1, 4) .$] #solutionbox[ #math.equation(block: true, alt: "x equals −2")[$x = −2$] ] ] Notice in the previous example, once we cleared the equation of fractions, the equation was like those we solved earlier in this chapter. We changed the problem to one we already knew how to solve. We then used the #strong[General Strategy for Solving Linear Equations]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Solve Equations with Fraction or Decimal Coefficients.] + Find the least common denominator (LCD) of #emph[all] the fractions and decimals (in fraction form) in the equation. + Multiply both sides of the equation by that LCD. This clears the fractions and decimals. + Solve using the General Strategy for Solving Linear Equations. ] #examplebox("Example 10")[][ Solve: #math.equation(block: false, alt: "5 equals the fraction 1 over 2 y plus the fraction 2 over 3 y minus the fraction 3 over 4 y .")[$5 = frac(1, 2) y + frac(2, 3) y − frac(3, 4) y .$] #solutionbox[ We want to clear the fractions by multiplying both sides of the equation by the LCD of all the fractions in the equation. #figure(table( columns: 3, align: left, inset: 6pt, [Find the LCD of all fractions in the equation.], [#figure(figph[A mathematical equation is displayed on a white background: 5 = 1/2y + 2/3y - 3/4y.], alt: "A mathematical equation is displayed on a white background: 5 = 1/2y + 2/3y - 3/4y.", caption: none)], [], [The LCD is 12.], [], [], [Multiply both sides of the equation by 12.], [#figure(figph[A mathematical equation shown as 12(5) = 12 \* (1/2y + 2/3y - 3/4y). The numbers 12 are highlighted in red, indicating a multiplication on both sides of the equation.], alt: "A mathematical equation shown as 12(5) = 12 * (1/2y + 2/3y - 3/4y). The numbers 12 are highlighted in red, indicating a multiplication on both sides of the equation.", caption: none)], [], [Distribute.], [#figure(figph[An algebraic equation showing 12 multiplied by 5 on the left side, equal to 12 times one-half y, plus 12 times two-thirds y, minus 12 times three-quarters y on the right side.], alt: "An algebraic equation showing 12 multiplied by 5 on the left side, equal to 12 times one-half y, plus 12 times two-thirds y, minus 12 times three-quarters y on the right side.", caption: none)], [], [Simplify—notice, no more fractions.], [#figure(figph[A mathematical equation on a white background reads '60 = 6y + 8y - 9y'.], alt: "A mathematical equation on a white background reads '60 = 6y + 8y - 9y'.", caption: none)], [], [Combine like terms.], [#figure(figph[A clear image displaying the algebraic equation 60 = 5y. This simple linear equation can be solved by dividing both sides by 5 to find the value of y, which would be 12.], alt: "A clear image displaying the algebraic equation 60 = 5y. This simple linear equation can be solved by dividing both sides by 5 to find the value of y, which would be 12.", caption: none)], [], [Divide by five.], [#figure(figph[A mathematical equation shows '60 divided by 5 equals 5y divided by 5'. The number 5 in the denominators is highlighted in red.], alt: "A mathematical equation shows '60 divided by 5 equals 5y divided by 5'. The number 5 in the denominators is highlighted in red.", caption: none)], [], [Simplify.], [#figure(figph[A mathematical equation is displayed on a white background, showing the number '12' equal to the variable 'y', rendered in a simple, sans-serif font.], alt: "A mathematical equation is displayed on a white background, showing the number '12' equal to the variable 'y', rendered in a simple, sans-serif font.", caption: none)], [], [Check:], [#figure(figph[A mathematical equation is displayed, showing 5 equals one-half y plus two-thirds y minus three-fourths y. The equation involves fractions and a variable 'y'.], alt: "A mathematical equation is displayed, showing 5 equals one-half y plus two-thirds y minus three-fourths y. The equation involves fractions and a variable 'y'.", caption: none)], [], [Let #math.equation(block: false, alt: "y equals 12.")[$y = 12.$]], [#figure(figph[A mathematical equation questions whether 5 is equal to the expression: one-half of 12, plus two-thirds of 12, minus three-fourths of 12, with the number 12 highlighted in red.], alt: "A mathematical equation questions whether 5 is equal to the expression: one-half of 12, plus two-thirds of 12, minus three-fourths of 12, with the number 12 highlighted in red.", caption: none)], [], [], [#figure(figph[A mathematical expression 5 =? 6 + 8 - 9 is displayed, asking if 5 is equal to the result of 6 plus 8 minus 9.], alt: "A mathematical expression 5 =? 6 + 8 - 9 is displayed, asking if 5 is equal to the result of 6 plus 8 minus 9.", caption: none)], [], [], [#figure(figph[The equation 5 = 5 with a checkmark to its right, indicating that the equality is correct. The image displays numerical equality and verification on a white background.], alt: "The equation 5 = 5 with a checkmark to its right, indicating that the equality is correct. The image displays numerical equality and verification on a white background.", caption: none)], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "7 equals the fraction 1 over 2 x plus the fraction 3 over 4 x minus the fraction 2 over 3 x .")[$7 = frac(1, 2) x + frac(3, 4) x − frac(2, 3) x .$] #solutionbox[ #math.equation(block: true, alt: "x equals 12")[$x = 12$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "−1 equals the fraction 1 over 2 u plus the fraction 1 over 4 u minus the fraction 2 over 3 u .")[$−1 = frac(1, 2) u + frac(1, 4) u − frac(2, 3) u .$] #solutionbox[ #math.equation(block: true, alt: "u equals −12")[$u = −12$] ] ] In the next example, we’ll distribute before we clear the fractions. #examplebox("Example 11")[][ Solve: #math.equation(block: false, alt: "the fraction 1 over 2 open parenthesis y minus 5 close parenthesis equals the fraction 1 over 4 open parenthesis y minus 1 close parenthesis .")[$frac(1, 2) ( y − 5 ) = frac(1, 4) ( y − 1 ) .$] #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, [], [#figure(figph[A mathematical equation is shown: one-half times the quantity y minus 5 equals one-fourth times the quantity y minus 1. This can also be written as (1/2)(y-5)=(1/4)(y-1).], alt: "A mathematical equation is shown: one-half times the quantity y minus 5 equals one-fourth times the quantity y minus 1. This can also be written as (1/2)(y-5)=(1/4)(y-1).", caption: none)], [], [Distribute.], [#figure(figph[A mathematical equation is displayed: (1/2) \* y - (1/2) \* 5 = (1/4) \* y - (1/4) \* 1. This algebraic expression involves fractions, multiplication, subtraction, and the variable 'y'.], alt: "A mathematical equation is displayed: (1/2) * y - (1/2) * 5 = (1/4) * y - (1/4) * 1. This algebraic expression involves fractions, multiplication, subtraction, and the variable 'y'.", caption: none)], [], [Simplify.], [#figure(figph[A mathematical equation is displayed: 1/2y - 5/2 = 1/4y - 1/4.], alt: "A mathematical equation is displayed: 1/2y - 5/2 = 1/4y - 1/4.", caption: none)], [], [Multiply by the LCD, four.], [#figure(figph[An equation showing 4(1/2y - 5/2) = 4(1/4y - 1/4). The number 4 is highlighted in red on both sides of the equation.], alt: "An equation showing 4(1/2y - 5/2) = 4(1/4y - 1/4). The number 4 is highlighted in red on both sides of the equation.", caption: none)], [], [Distribute.], [#figure(figph[A mathematical equation shows '4 multiplied by 1/2y minus 4 multiplied by 5/2 equals 4 multiplied by 1/4y minus 4 multiplied by 1/4'.], alt: "A mathematical equation shows '4 multiplied by 1/2y minus 4 multiplied by 5/2 equals 4 multiplied by 1/4y minus 4 multiplied by 1/4'.", caption: none)], [], [Simplify.], [#figure(figph[A mathematical equation is displayed, reading '2y - 10 = y - 1'.], alt: "A mathematical equation is displayed, reading '2y - 10 = y - 1'.", caption: none)], [], [Collect the variables to the left.], [#figure(figph[A mathematical equation is displayed, showing '2y - y - 10 = y - y - 1'. The variable 'y' is shown in both black and red, while constants and operators are in black.], alt: "A mathematical equation is displayed, showing '2y - y - 10 = y - y - 1'. The variable 'y' is shown in both black and red, while constants and operators are in black.", caption: none)], [], [Simplify.], [#figure(figph[A mathematical equation is displayed, showing 'y - 10 = -1' in a clear, dark font against a plain white background.], alt: "A mathematical equation is displayed, showing 'y - 10 = -1' in a clear, dark font against a plain white background.", caption: none)], [], [Collect the constants to the right.], [#figure(figph[The equation y - 10 + 10 = -1 + 10 shows a step in solving for y by adding 10 to both sides, effectively canceling out the -10 on the left to isolate y.], alt: "The equation y - 10 + 10 = -1 + 10 shows a step in solving for y by adding 10 to both sides, effectively canceling out the -10 on the left to isolate y.", caption: none)], [], [Simplify.], [#figure(figph[A simple mathematical equation 'y = 9' is displayed in a clear, dark gray font on a plain white background.], alt: "A simple mathematical equation 'y = 9' is displayed in a clear, dark gray font on a plain white background.", caption: none)], [], [An alternate way to solve this equation is to clear the fractions without distributing first. If you multiply the factors correctly, this method will be easier.], [], [], [], [#figure(figph[A mathematical equation is displayed, showing 'one half (y minus five) equals one fourth (y minus one)', representing a linear equation to be solved for the variable y.], alt: "A mathematical equation is displayed, showing 'one half (y minus five) equals one fourth (y minus one)', representing a linear equation to be solved for the variable y.", caption: none)], [], [Multiply by the LCD, 4.], [#figure(figph[A mathematical equation displays the multiplication of 4 by 1/2(y-5) on the left side, and 4 by 1/4(y-1) on the right side, with the number 4 highlighted in red.], alt: "A mathematical equation displays the multiplication of 4 by 1/2(y-5) on the left side, and 4 by 1/4(y-1) on the right side, with the number 4 highlighted in red.", caption: none)], [], [Multiply four times the fractions.], [#figure(figph[A mathematical equation is displayed on a white background: 2(y - 5) = 1(y - 1).], alt: "A mathematical equation is displayed on a white background: 2(y - 5) = 1(y - 1).", caption: none)], [], [Distribute.], [#figure(figph[A mathematical equation is displayed, reading '2y - 10 = y - 1' in a clear, dark gray font against a plain white background.], alt: "A mathematical equation is displayed, reading '2y - 10 = y - 1' in a clear, dark gray font against a plain white background.", caption: none)], [], [Collect the variables to the left.], [#figure(figph[A mathematical equation, 2y - y - 10 = y - y - 1, is displayed on a white background. Some 'y' terms are colored red, likely indicating subtraction or grouping of like terms in an algebra problem.], alt: "A mathematical equation, 2y - y - 10 = y - y - 1, is displayed on a white background. Some 'y' terms are colored red, likely indicating subtraction or grouping of like terms in an algebra problem.", caption: none)], [], [Simplify.], [#figure(figph[A mathematical equation is displayed, reading 'y - 10 = -1'.], alt: "A mathematical equation is displayed, reading 'y - 10 = -1'.", caption: none)], [], [Collect the constants to the right.], [#figure(figph[A mathematical equation is displayed on a white background, showing 'y - 10 + 10 = -1 + 10' with the numbers '+ 10' on both sides highlighted in red.], alt: "A mathematical equation is displayed on a white background, showing 'y - 10 + 10 = -1 + 10' with the numbers '+ 10' on both sides highlighted in red.", caption: none)], [], [Simplify.], [#figure(figph[The image displays the equation 'y = 9' in the center of a white background. The text is rendered in a clear, dark gray font.], alt: "The image displays the equation 'y = 9' in the center of a white background. The text is rendered in a clear, dark gray font.", caption: none)], [], [Check:], [#figure(figph[A mathematical equation is displayed: 1/2(y - 5) = 1/4(y - 1).], alt: "A mathematical equation is displayed: 1/2(y - 5) = 1/4(y - 1).", caption: none)], [], [Let #math.equation(block: false, alt: "y equals 9.")[$y = 9.$]], [#figure(figph[A mathematical equation asking if 1/2(9-5) equals 1/4(9-1). The numbers 9 and 5 in the first parenthesis are highlighted in red, as are 9 and 1 in the second parenthesis.], alt: "A mathematical equation asking if 1/2(9-5) equals 1/4(9-1). The numbers 9 and 5 in the first parenthesis are highlighted in red, as are 9 and 1 in the second parenthesis.", caption: none)], [], [Finish the check on your own.], [], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction 1 over 5 open parenthesis n plus 3 close parenthesis equals the fraction 1 over 4 open parenthesis n plus 2 close parenthesis .")[$frac(1, 5) ( n + 3 ) = frac(1, 4) ( n + 2 ) .$] #solutionbox[ #math.equation(block: true, alt: "n equals 2")[$n = 2$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction 1 over 2 open parenthesis m minus 3 close parenthesis equals the fraction 1 over 4 open parenthesis m minus 7 close parenthesis .")[$frac(1, 2) ( m − 3 ) = frac(1, 4) ( m − 7 ) .$] #solutionbox[ #math.equation(block: true, alt: "m equals −1")[$m = −1$] ] ] When you multiply both sides of an equation by the LCD of the fractions, make sure you multiply each term by the LCD—even if it does not contain a fraction. #examplebox("Example 12")[][ Solve: #math.equation(block: false, alt: "the fraction 4 q plus 3 over 2 plus 6 equals the fraction 3 q plus 5 over 4")[$frac(4 q + 3, 2) + 6 = frac(3 q + 5, 4)$] #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, [], [#figure(figph[A mathematical equation is shown: (4q + 3) / 2 + 6 = (3q + 5) / 4. It is an algebraic expression involving the variable 'q' and fractions.], alt: "A mathematical equation is shown: (4q + 3) / 2 + 6 = (3q + 5) / 4. It is an algebraic expression involving the variable 'q' and fractions.", caption: none)], [], [Multiply both sides by the LCD, 4.], [#figure(figph[A mathematical equation shows '4 multiplied by the quantity (4q+3)/2 plus 6', which is equal to '4 multiplied by the quantity (3q+5)/4'.], alt: "A mathematical equation shows '4 multiplied by the quantity (4q+3)/2 plus 6', which is equal to '4 multiplied by the quantity (3q+5)/4'.", caption: none)], [], [Distribute.], [#figure(figph[An algebraic equation: 4 multiplied by the fraction (4q + 3) / 2, plus 4 multiplied by 6, equals 4 multiplied by the fraction (3q + 5) / 4.], alt: "An algebraic equation: 4 multiplied by the fraction (4q + 3) / 2, plus 4 multiplied by 6, equals 4 multiplied by the fraction (3q + 5) / 4.", caption: none)], [], [Simplify.], [#figure(figph[A mathematical equation is displayed on a white background: 2(4q + 3) + 24 = 3q + 5.], alt: "A mathematical equation is displayed on a white background: 2(4q + 3) + 24 = 3q + 5.", caption: none)], [], [], [#figure(figph[An image displays the algebraic equation 8q + 6 + 24 = 3q + 5 in black text on a white background. The equation involves a variable 'q' and various integer constants on both sides of the equality.], alt: "An image displays the algebraic equation 8q + 6 + 24 = 3q + 5 in black text on a white background. The equation involves a variable 'q' and various integer constants on both sides of the equality.", caption: none)], [], [], [#figure(figph[A mathematical equation is displayed: 8q + 30 = 3q + 5.], alt: "A mathematical equation is displayed: 8q + 30 = 3q + 5.", caption: none)], [], [Collect the variables to the left.], [#figure(figph[A mathematical equation is displayed, showing '8q - 3q + 30 = 3q - 3q + 5'. Some 'q' terms are in black, while others are in red.], alt: "A mathematical equation is displayed, showing '8q - 3q + 30 = 3q - 3q + 5'. Some 'q' terms are in black, while others are in red.", caption: none)], [], [Simplify.], [#figure(figph[A white background displays the mathematical equation 5q + 30 = 5, rendered in clear, dark gray text.], alt: "A white background displays the mathematical equation 5q + 30 = 5, rendered in clear, dark gray text.", caption: none)], [], [Collect the constants to the right.], [#figure(figph[An algebra equation 5q + 30 - 30 = 5 - 30, demonstrating the step of subtracting 30 from both sides to isolate the variable, with the subtracted 30s highlighted in red.], alt: "An algebra equation 5q + 30 - 30 = 5 - 30, demonstrating the step of subtracting 30 from both sides to isolate the variable, with the subtracted 30s highlighted in red.", caption: none)], [], [Simplify.], [#figure(figph[A mathematical equation is displayed on a white background, showing '5q = -25' in black text.], alt: "A mathematical equation is displayed on a white background, showing '5q = -25' in black text.", caption: none)], [], [Divide both sides by five.], [#figure(figph[Solving for q: Both sides of the equation 5q = -25 are divided by 5, resulting in 5q/5 = -25/5 to find the value of q.], alt: "Solving for q: Both sides of the equation 5q = -25 are divided by 5, resulting in 5q/5 = -25/5 to find the value of q.", caption: none)], [], [Simplify.], [#figure(figph[The image displays the mathematical equation 'q = -5' in black text against a plain white background, presenting a simple assignment of the value negative five to the variable q.], alt: "The image displays the mathematical equation 'q = -5' in black text against a plain white background, presenting a simple assignment of the value negative five to the variable q.", caption: none)], [], [Check:], [#figure(figph[A mathematical equation is displayed: (4q + 3) / 2 + 6 = (3q + 5) / 4.], alt: "A mathematical equation is displayed: (4q + 3) / 2 + 6 = (3q + 5) / 4.", caption: none)], [], [Let #math.equation(block: false, alt: "q equals minus 5.")[$q = − 5.$]], [#figure(figph[A mathematical equation asking to compare two expressions: (4(-5)+3)/2+6 on the left, and (3(-5)+5)/4 on the right, separated by a question mark and an equals sign.], alt: "A mathematical equation asking to compare two expressions: (4(-5)+3)/2+6 on the left, and (3(-5)+5)/4 on the right, separated by a question mark and an equals sign.", caption: none)], [], [Finish the check on your own.], [], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction 3 r plus 5 over 6 plus 1 equals the fraction 4 r plus 3 over 3 .")[$frac(3 r + 5, 6) + 1 = frac(4 r + 3, 3) .$] #solutionbox[ #math.equation(block: true, alt: "r equals 1")[$r = 1$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction 2 s plus 3 over 2 plus 1 equals the fraction 3 s plus 2 over 4 .")[$frac(2 s + 3, 2) + 1 = frac(3 s + 2, 4) .$] #solutionbox[ #math.equation(block: true, alt: "s equals −8")[$s = −8$] ] ] Some equations have decimals in them. This kind of equation may occur when we solve problems dealing with money or percentages. But decimals can also be expressed as fractions. For example, #math.equation(block: false, alt: "0.7 equals the fraction 7 over 10")[$0.7 = frac(7, 10)$] and #math.equation(block: false, alt: "0.29 equals the fraction 29 over 100 .")[$0.29 = frac(29, 100) .$] So, with an equation with decimals, we can use the same method we used to clear fractions—multiply both sides of the equation by the #strong[least common denominator]. The next example uses an equation that is typical of the ones we will see in the money applications in a later section. Notice that we will clear all decimals by multiplying by the LCD of their fraction form. #examplebox("Example 13")[][ Solve: #math.equation(block: false, alt: "0.25 x plus 0.05 open parenthesis x plus 3 close parenthesis equals 2.85 .")[$0.25 x + 0.05 ( x + 3 ) = 2.85 .$] #solutionbox[ Look at the decimals and think of the equivalent fractions: #math.equation(block: true, alt: "0.25 equals the fraction 25 over 100 , 0.05 equals the fraction 5 over 100 , 2.85 equals 2 the fraction 85 over 100 .")[$0.25 = frac(25, 100) , #h(3em) 0.05 = frac(5, 100) , #h(3em) 2.85 = 2 frac(85, 100) .$]Notice, the LCD is 100. By multiplying by the LCD we will clear the decimals from the equation. #figure(table( columns: 2, align: left, inset: 6pt, [], [#figure(figph[A mathematical equation is displayed, reading 0.25x + 0.05(x + 3) = 2.85.], alt: "A mathematical equation is displayed, reading 0.25x + 0.05(x + 3) = 2.85.", caption: none)], [Distribute first.], [#figure(figph[A mathematical equation is displayed with the expression 0.25x + 0.05x + 0.15 = 2.85.], alt: "A mathematical equation is displayed with the expression 0.25x + 0.05x + 0.15 = 2.85.", caption: none)], [Combine like terms.], [#figure(figph[A mathematical equation is displayed, reading '0.30x + 0.15 = 2.85' against a white background.], alt: "A mathematical equation is displayed, reading '0.30x + 0.15 = 2.85' against a white background.", caption: none)], [To clear decimals, multiply by 100.], [#figure(figph[A mathematical equation is displayed, showing 100 multiplied by the quantity (0.30x + 0.15) on the left side, which equals 100 multiplied by 2.85 on the right side.], alt: "A mathematical equation is displayed, showing 100 multiplied by the quantity (0.30x + 0.15) on the left side, which equals 100 multiplied by 2.85 on the right side.", caption: none)], [Distribute.], [#figure(figph[A mathematical equation is displayed, showing 30x + 15 = 285.], alt: "A mathematical equation is displayed, showing 30x + 15 = 285.", caption: none)], [Subtract 15 from both sides.], [#figure(figph[An algebraic equation showing the step of subtracting 15 from both sides: 30x + 15 - 15 = 285 - 15.], alt: "An algebraic equation showing the step of subtracting 15 from both sides: 30x + 15 - 15 = 285 - 15.", caption: none)], [Simplify.], [#figure(figph[A mathematical equation shows '30x = 270' against a white background.], alt: "A mathematical equation shows '30x = 270' against a white background.", caption: none)], [Divide by 30.], [#figure(figph[The equation 30x/30 = 270/30 illustrates a step in solving for 'x' where both sides are divided by 30 to isolate the variable, leading to x = 9.], alt: "The equation 30x/30 = 270/30 illustrates a step in solving for 'x' where both sides are divided by 30 to isolate the variable, leading to x = 9.", caption: none)], [Simplify.], [#figure(figph[The mathematical equation 'x=9' is displayed on a plain white background, presenting a simple statement of equality.], alt: "The mathematical equation 'x=9' is displayed on a plain white background, presenting a simple statement of equality.", caption: none)], [Check it yourself by substituting #math.equation(block: false, alt: "x equals 9")[$x = 9$] into the original equation.], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "0.25 n plus 0.05 open parenthesis n plus 5 close parenthesis equals 2.95 .")[$0.25 n + 0.05 ( n + 5 ) = 2.95 .$] #solutionbox[ #math.equation(block: true, alt: "n equals 9")[$n = 9$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "0.10 d plus 0.05 open parenthesis d minus 5 close parenthesis equals 2.15 .")[$0.10 d + 0.05 ( d − 5 ) = 2.15 .$] #solutionbox[ #math.equation(block: true, alt: "d equals 16")[$d = 16$] ] ] === Key Concepts - #strong[How to determine whether a number is a solution to an equation] + Substitute the number in for the variable in the equation. + Simplify the expressions on both sides of the equation. + Determine whether the resulting equation is true. #linebreak() If it is true, the number is a solution. #linebreak() If it is not true, the number is not a solution. - #strong[How to Solve Linear Equations Using a General Strategy] + Simplify each side of the equation as much as possible. #linebreak() Use the Distributive Property to remove any parentheses. #linebreak() Combine like terms. + Collect all the variable terms on one side of the equation. #linebreak() Use the Addition or Subtraction Property of Equality. + Collect all the constant terms on the other side of the equation. #linebreak() Use the Addition or Subtraction Property of Equality. + Make the coefficient of the variable term equal to 1. #linebreak() Use the Multiplication or Division Property of Equality. #linebreak() State the solution to the equation. + Check the solution. #linebreak() Substitute the solution into the original equation to make sure the result is a true statement. - #strong[How to Solve Equations with Fraction or Decimal Coefficients] + Find the least common denominator (LCD) of #emph[all] the fractions and decimals (in fraction form) in the equation. + Multiply both sides of the equation by that LCD. This clears the fractions and decimals. + Solve using the General Strategy for Solving Linear Equations. ==== Practice Makes Perfect #strong[Solve Equations Using the General Strategy] In the following exercises, determine whether the given values are solutions to the equation. #math.equation(block: true, alt: "6 y plus 10 equals 12 y")[$6 y + 10 = 12 y$] ⓐ #math.equation(block: false, alt: "y equals the fraction 5 over 3")[$y = frac(5, 3)$] ⓑ #math.equation(block: false, alt: "y equals minus the fraction 1 over 2")[$y = − frac(1, 2)$] #solutionbox[ ⓐ yes ⓑ no ] #math.equation(block: true, alt: "4 x plus 9 equals 8 x")[$4 x + 9 = 8 x$] ⓐ #math.equation(block: false, alt: "x equals minus the fraction 7 over 8")[$x = − frac(7, 8)$] ⓑ #math.equation(block: false, alt: "x equals the fraction 9 over 4")[$x = frac(9, 4)$] #math.equation(block: true, alt: "8 u minus 1 equals 6 u")[$8 u − 1 = 6 u$] ⓐ #math.equation(block: false, alt: "u equals minus the fraction 1 over 2")[$u = − frac(1, 2)$] ⓑ #math.equation(block: false, alt: "u equals the fraction 1 over 2")[$u = frac(1, 2)$] #solutionbox[ ⓐ no ⓑ yes ] #math.equation(block: true, alt: "9 v minus 2 equals 3 v")[$9 v − 2 = 3 v$] ⓐ #math.equation(block: false, alt: "v equals minus the fraction 1 over 3")[$v = − frac(1, 3)$] ⓑ #math.equation(block: false, alt: "v equals the fraction 1 over 3")[$v = frac(1, 3)$] In the following exercises, solve each linear equation. #math.equation(block: true, alt: "15 open parenthesis y minus 9 close parenthesis equals −60")[$15 ( y − 9 ) = −60$] #solutionbox[ #math.equation(block: true, alt: "y equals 5")[$y = 5$] ] #math.equation(block: true, alt: "−16 open parenthesis 3 n plus 4 close parenthesis equals 32")[$−16 ( 3 n + 4 ) = 32$] #math.equation(block: true, alt: "− open parenthesis w minus 12 close parenthesis equals 30")[$"−" ( w − 12 ) = 30$] #solutionbox[ #math.equation(block: true, alt: "w equals −18")[$w = −18$] ] #math.equation(block: true, alt: "− open parenthesis t minus 19 close parenthesis equals 28")[$"−" ( t − 19 ) = 28$] #math.equation(block: true, alt: "51 plus 5 open parenthesis 4 minus q close parenthesis equals 56")[$51 + 5 ( 4 − q ) = 56$] #solutionbox[ #math.equation(block: true, alt: "q equals 3")[$q = 3$] ] #math.equation(block: true, alt: "−6 plus 6 open parenthesis 5 minus k close parenthesis equals 15")[$−6 + 6 ( 5 − k ) = 15$] #math.equation(block: true, alt: "3 open parenthesis 10 minus 2 x close parenthesis plus 54 equals 0")[$3 ( 10 − 2 x ) + 54 = 0$] #solutionbox[ #math.equation(block: true, alt: "x equals 14")[$x = 14$] ] #math.equation(block: true, alt: "−2 open parenthesis 11 minus 7 x close parenthesis plus 54 equals 4")[$−2 ( 11 − 7 x ) + 54 = 4$] #math.equation(block: true, alt: "the fraction 2 over 3 open parenthesis 9 c minus 3 close parenthesis equals 22")[$frac(2, 3) ( 9 c − 3 ) = 22$] #solutionbox[ #math.equation(block: true, alt: "c equals 4")[$c = 4$] ] #math.equation(block: true, alt: "the fraction 3 over 5 open parenthesis 10 x minus 5 close parenthesis equals 27")[$frac(3, 5) ( 10 x − 5 ) = 27$] #math.equation(block: true, alt: "the fraction 1 over 5 open parenthesis 15 c plus 10 close parenthesis equals c plus 7")[$frac(1, 5) ( 15 c + 10 ) = c + 7$] #solutionbox[ #math.equation(block: true, alt: "c equals the fraction 5 over 2")[$c = frac(5, 2)$] ] #math.equation(block: true, alt: "the fraction 1 over 4 open parenthesis 20 d plus 12 close parenthesis equals d plus 7")[$frac(1, 4) ( 20 d + 12 ) = d + 7$] #math.equation(block: true, alt: "3 open parenthesis 4 n minus 1 close parenthesis minus 2 equals 8 n plus 3")[$3 ( 4 n − 1 ) − 2 = 8 n + 3$] #solutionbox[ #math.equation(block: true, alt: "n equals 2")[$n = 2$] ] #math.equation(block: true, alt: "9 open parenthesis 2 m minus 3 close parenthesis minus 8 equals 4 m plus 7")[$9 ( 2 m − 3 ) − 8 = 4 m + 7$] #math.equation(block: true, alt: "12 plus 2 open parenthesis 5 minus 3 y close parenthesis equals −9 open parenthesis y minus 1 close parenthesis minus 2")[$12 + 2 ( 5 − 3 y ) = −9 ( y − 1 ) − 2$] #solutionbox[ #math.equation(block: true, alt: "y equals −5")[$y = −5$] ] #math.equation(block: true, alt: "−15 plus 4 open parenthesis 2 minus 5 y close parenthesis equals −7 open parenthesis y minus 4 close parenthesis plus 4")[$−15 + 4 ( 2 − 5 y ) = −7 ( y − 4 ) + 4$] #math.equation(block: true, alt: "5 plus 6 open parenthesis 3 s minus 5 close parenthesis equals −3 plus 2 open parenthesis 8 s minus 1 close parenthesis")[$5 + 6 ( 3 s − 5 ) = −3 + 2 ( 8 s − 1 )$] #solutionbox[ #math.equation(block: true, alt: "s equals 10")[$s = 10$] ] #math.equation(block: true, alt: "−12 plus 8 open parenthesis x minus 5 close parenthesis equals −4 plus 3 open parenthesis 5 x minus 2 close parenthesis")[$−12 + 8 ( x − 5 ) = −4 + 3 ( 5 x − 2 )$] #math.equation(block: true, alt: "4 open parenthesis p minus 4 close parenthesis minus open parenthesis p plus 7 close parenthesis equals 5 open parenthesis p minus 3 close parenthesis")[$4 ( p − 4 ) − ( p + 7 ) = 5 ( p − 3 )$] #solutionbox[ #math.equation(block: true, alt: "p equals −4")[$p = −4$] ] #math.equation(block: true, alt: "3 open parenthesis a minus 2 close parenthesis minus open parenthesis a plus 6 close parenthesis equals 4 open parenthesis a minus 1 close parenthesis")[$3 ( a − 2 ) − ( a + 6 ) = 4 ( a − 1 )$] #math.equation(block: true, alt: "4 [ 5 minus 8 open parenthesis 4 c minus 3 close parenthesis ] equals 12 open parenthesis 1 minus 13 c close parenthesis minus 8")[$4 [ 5 − 8 ( 4 c − 3 ) ] = 12 ( 1 − 13 c ) − 8$] #solutionbox[ #math.equation(block: true, alt: "c equals −4")[$c = −4$] ] #math.equation(block: true, alt: "5 [ 9 minus 2 open parenthesis 6 d minus 1 close parenthesis ] equals 11 open parenthesis 4 minus 10 d close parenthesis minus 139")[$5 [ 9 − 2 ( 6 d − 1 ) ] = 11 ( 4 − 10 d ) − 139$] #math.equation(block: true, alt: "3 [ −9 plus 8 open parenthesis 4 h minus 3 close parenthesis ] equals 2 open parenthesis 5 minus 12 h close parenthesis minus 19")[$3 [ −9 + 8 ( 4 h − 3 ) ] = 2 ( 5 − 12 h ) − 19$] #solutionbox[ #math.equation(block: true, alt: "h equals the fraction 3 over 4")[$h = frac(3, 4)$] ] #math.equation(block: true, alt: "3 [ −14 plus 2 open parenthesis 15 k minus 6 close parenthesis ] equals 8 open parenthesis 3 minus 5 k close parenthesis minus 24")[$3 [ −14 + 2 ( 15 k − 6 ) ] = 8 ( 3 − 5 k ) − 24$] #math.equation(block: true, alt: "5 [ 2 open parenthesis m plus 4 close parenthesis plus 8 open parenthesis m minus 7 close parenthesis ] equals 2 [ 3 open parenthesis 5 plus m close parenthesis minus open parenthesis 21 minus 3 m close parenthesis ]")[$5 [ 2 ( m + 4 ) + 8 ( m − 7 ) ] = 2 [ 3 ( 5 + m ) − ( 21 − 3 m ) ]$] #solutionbox[ #math.equation(block: true, alt: "m equals 6")[$m = 6$] ] #math.equation(block: true, alt: "10 [ 5 open parenthesis n plus 1 close parenthesis plus 4 open parenthesis n minus 1 close parenthesis ] equals 11 [ 7 open parenthesis 5 plus n close parenthesis minus open parenthesis 25 minus 3 n close parenthesis ]")[$10 [ 5 ( n + 1 ) + 4 ( n − 1 ) ] = 11 [ 7 ( 5 + n ) − ( 25 − 3 n ) ]$] #strong[Classify Equations] In the following exercises, classify each equation as a conditional equation, an identity, or a contradiction and then state the solution. #math.equation(block: true, alt: "23 z plus 19 equals 3 open parenthesis 5 z minus 9 close parenthesis plus 8 z plus 46")[$23 z + 19 = 3 ( 5 z − 9 ) + 8 z + 46$] #solutionbox[ identity; all real numbers ] #math.equation(block: true, alt: "15 y plus 32 equals 2 open parenthesis 10 y minus 7 close parenthesis minus 5 y plus 46")[$15 y + 32 = 2 ( 10 y − 7 ) − 5 y + 46$] #math.equation(block: true, alt: "18 open parenthesis 5 j minus 1 close parenthesis plus 29 equals 47")[$18 ( 5 j − 1 ) + 29 = 47$] #solutionbox[ conditional equation;#math.equation(block: false, alt: "j equals the fraction 2 over 5")[$j = frac(2, 5)$] ] #math.equation(block: true, alt: "24 open parenthesis 3 d minus 4 close parenthesis plus 100 equals 52")[$24 ( 3 d − 4 ) + 100 = 52$] #math.equation(block: true, alt: "22 open parenthesis 3 m minus 4 close parenthesis equals 8 open parenthesis 2 m plus 9 close parenthesis")[$22 ( 3 m − 4 ) = 8 ( 2 m + 9 )$] #solutionbox[ conditional equation; #math.equation(block: false, alt: "m equals the fraction 16 over 5")[$m = frac(16, 5)$] ] #math.equation(block: true, alt: "30 open parenthesis 2 n minus 1 close parenthesis equals 5 open parenthesis 10 n plus 8 close parenthesis")[$30 ( 2 n − 1 ) = 5 ( 10 n + 8 )$] #math.equation(block: true, alt: "7 v plus 42 equals 11 open parenthesis 3 v plus 8 close parenthesis minus 2 open parenthesis 13 v minus 1 close parenthesis")[$7 v + 42 = 11 ( 3 v + 8 ) − 2 ( 13 v − 1 )$] #solutionbox[ contradiction; no solution ] #math.equation(block: true, alt: "18 u minus 51 equals 9 open parenthesis 4 u plus 5 close parenthesis minus 6 open parenthesis 3 u minus 10 close parenthesis")[$18 u − 51 = 9 ( 4 u + 5 ) − 6 ( 3 u − 10 )$] #math.equation(block: true, alt: "45 open parenthesis 3 y minus 2 close parenthesis equals 9 open parenthesis 15 y minus 6 close parenthesis")[$45 ( 3 y − 2 ) = 9 ( 15 y − 6 )$] #solutionbox[ contradiction; no solution ] #math.equation(block: true, alt: "60 open parenthesis 2 x minus 1 close parenthesis equals 15 open parenthesis 8 x plus 5 close parenthesis")[$60 ( 2 x − 1 ) = 15 ( 8 x + 5 )$] #math.equation(block: true, alt: "9 open parenthesis 14 d plus 9 close parenthesis plus 4 d equals 13 open parenthesis 10 d plus 6 close parenthesis plus 3")[$9 ( 14 d + 9 ) + 4 d = 13 ( 10 d + 6 ) + 3$] #solutionbox[ identity; all real numbers ] #math.equation(block: true, alt: "11 open parenthesis 8 c plus 5 close parenthesis minus 8 c equals 2 open parenthesis 40 c plus 25 close parenthesis plus 5")[$11 ( 8 c + 5 ) − 8 c = 2 ( 40 c + 25 ) + 5$] #strong[Solve Equations with Fraction or Decimal Coefficients] In the following exercises, solve each equation with fraction coefficients. #math.equation(block: true, alt: "the fraction 1 over 4 x minus the fraction 1 over 2 equals minus the fraction 3 over 4")[$frac(1, 4) x − frac(1, 2) = − frac(3, 4)$] #solutionbox[ #math.equation(block: true, alt: "x equals −1")[$x = −1$] ] #math.equation(block: true, alt: "the fraction 3 over 4 x minus the fraction 1 over 2 equals the fraction 1 over 4")[$frac(3, 4) x − frac(1, 2) = frac(1, 4)$] #math.equation(block: true, alt: "the fraction 5 over 6 y minus the fraction 2 over 3 equals minus the fraction 3 over 2")[$frac(5, 6) y − frac(2, 3) = − frac(3, 2)$] #solutionbox[ #math.equation(block: true, alt: "y equals −1")[$y = −1$] ] #math.equation(block: true, alt: "the fraction 5 over 6 y minus the fraction 1 over 3 equals minus the fraction 7 over 6")[$frac(5, 6) y − frac(1, 3) = − frac(7, 6)$] #math.equation(block: true, alt: "the fraction 1 over 2 a plus the fraction 3 over 8 equals the fraction 3 over 4")[$frac(1, 2) a + frac(3, 8) = frac(3, 4)$] #solutionbox[ #math.equation(block: true, alt: "a equals the fraction 3 over 4")[$a = frac(3, 4)$] ] #math.equation(block: true, alt: "the fraction 5 over 8 b plus the fraction 1 over 2 equals minus the fraction 3 over 4")[$frac(5, 8) b + frac(1, 2) = − frac(3, 4)$] #math.equation(block: true, alt: "2 equals the fraction 1 over 3 x minus the fraction 1 over 2 x plus the fraction 2 over 3 x")[$2 = frac(1, 3) x − frac(1, 2) x + frac(2, 3) x$] #solutionbox[ #math.equation(block: true, alt: "x equals 4")[$x = 4$] ] #math.equation(block: true, alt: "2 equals the fraction 3 over 5 x minus the fraction 1 over 3 x plus the fraction 2 over 5 x")[$2 = frac(3, 5) x − frac(1, 3) x + frac(2, 5) x$] #math.equation(block: true, alt: "the fraction 1 over 3 w plus the fraction 5 over 4 equals w minus the fraction 1 over 4")[$frac(1, 3) w + frac(5, 4) = w − frac(1, 4)$] #solutionbox[ #math.equation(block: true, alt: "w equals the fraction 9 over 4")[$w = frac(9, 4)$] ] #math.equation(block: true, alt: "the fraction 1 over 2 a minus the fraction 1 over 4 equals the fraction 1 over 6 a plus the fraction 1 over 12")[$frac(1, 2) a − frac(1, 4) = frac(1, 6) a + frac(1, 12)$] #math.equation(block: true, alt: "the fraction 1 over 3 b plus the fraction 1 over 5 equals the fraction 2 over 5 b minus the fraction 3 over 5")[$frac(1, 3) b + frac(1, 5) = frac(2, 5) b − frac(3, 5)$] #solutionbox[ #math.equation(block: true, alt: "b equals 12")[$b = 12$] ] #math.equation(block: true, alt: "the fraction 1 over 3 x plus the fraction 2 over 5 equals the fraction 1 over 5 x minus the fraction 2 over 5")[$frac(1, 3) x + frac(2, 5) = frac(1, 5) x − frac(2, 5)$] #math.equation(block: true, alt: "the fraction 1 over 4 open parenthesis p minus 7 close parenthesis equals the fraction 1 over 3 open parenthesis p plus 5 close parenthesis")[$frac(1, 4) ( p − 7 ) = frac(1, 3) ( p + 5 )$] #solutionbox[ #math.equation(block: true, alt: "p equals −41")[$p = −41$] ] #math.equation(block: true, alt: "the fraction 1 over 5 open parenthesis q plus 3 close parenthesis equals the fraction 1 over 2 open parenthesis q minus 3 close parenthesis")[$frac(1, 5) ( q + 3 ) = frac(1, 2) ( q − 3 )$] #math.equation(block: true, alt: "the fraction 1 over 2 open parenthesis x plus 4 close parenthesis equals the fraction 3 over 4")[$frac(1, 2) ( x + 4 ) = frac(3, 4)$] #solutionbox[ #math.equation(block: true, alt: "x equals minus the fraction 5 over 2")[$x = − frac(5, 2)$] ] #math.equation(block: true, alt: "the fraction 1 over 3 open parenthesis x plus 5 close parenthesis equals the fraction 5 over 6")[$frac(1, 3) ( x + 5 ) = frac(5, 6)$] #math.equation(block: true, alt: "the fraction 4 n plus 8 over 4 equals the fraction n over 3")[$frac(4 n + 8, 4) = frac(n, 3)$] #solutionbox[ #math.equation(block: true, alt: "n equals −3")[$n = −3$] ] #math.equation(block: true, alt: "the fraction 3 p plus 6 over 3 equals the fraction p over 2")[$frac(3 p + 6, 3) = frac(p, 2)$] #math.equation(block: true, alt: "the fraction 3 x plus 4 over 2 plus 1 equals the fraction 5 x plus 10 over 8")[$frac(3 x + 4, 2) + 1 = frac(5 x + 10, 8)$] #solutionbox[ #math.equation(block: true, alt: "x equals −2")[$x = −2$] ] #math.equation(block: true, alt: "the fraction 10 y minus 2 over 3 plus 3 equals the fraction 10 y plus 1 over 9")[$frac(10 y − 2, 3) + 3 = frac(10 y + 1, 9)$] #math.equation(block: true, alt: "the fraction 7 u minus 1 over 4 minus 1 equals the fraction 4 u plus 8 over 5")[$frac(7 u − 1, 4) − 1 = frac(4 u + 8, 5)$] #solutionbox[ #math.equation(block: true, alt: "u equals 3")[$u = 3$] ] #math.equation(block: true, alt: "the fraction 3 v minus 6 over 2 plus 5 equals the fraction 11 v minus 4 over 5")[$frac(3 v − 6, 2) + 5 = frac(11 v − 4, 5)$] In the following exercises, solve each equation with decimal coefficients. #math.equation(block: true, alt: "0.4 x plus 0.6 equals 0.5 x minus 1.2")[$0.4 x + 0.6 = 0.5 x − 1.2$] #solutionbox[ #math.equation(block: true, alt: "x equals 18")[$x = 18$] ] #math.equation(block: true, alt: "0.7 x plus 0.4 equals 0.6 x plus 2.4")[$0.7 x + 0.4 = 0.6 x + 2.4$] #math.equation(block: true, alt: "0.9 x minus 1.25 equals 0.75 x plus 1.75")[$0.9 x − 1.25 = 0.75 x + 1.75$] #solutionbox[ #math.equation(block: true, alt: "x equals 20")[$x = 20$] ] #math.equation(block: true, alt: "1.2 x minus 0.91 equals 0.8 x plus 2.29")[$1.2 x − 0.91 = 0.8 x + 2.29$] #math.equation(block: true, alt: "0.05 n plus 0.10 open parenthesis n plus 8 close parenthesis equals 2.15")[$0.05 n + 0.10 ( n + 8 ) = 2.15$] #solutionbox[ #math.equation(block: true, alt: "n equals 9")[$n = 9$] ] #math.equation(block: true, alt: "0.05 n plus 0.10 open parenthesis n plus 7 close parenthesis equals 3.55")[$0.05 n + 0.10 ( n + 7 ) = 3.55$] #math.equation(block: true, alt: "0.10 d plus 0.25 open parenthesis d plus 5 close parenthesis equals 4.05")[$0.10 d + 0.25 ( d + 5 ) = 4.05$] #solutionbox[ #math.equation(block: true, alt: "d equals 8")[$d = 8$] ] #math.equation(block: true, alt: "0.10 d plus 0.25 open parenthesis d plus 7 close parenthesis equals 5.25")[$0.10 d + 0.25 ( d + 7 ) = 5.25$] ==== Everyday Math #strong[Fencing] Micah has 74 feet of fencing to make a dog run in his yard. He wants the length to be 2.5 feet more than the width. Find the length, #emph[L], by solving the equation #math.equation(block: false, alt: "2 L plus 2 open parenthesis L minus 2.5 close parenthesis equals 74 .")[$2 L + 2 ( L − 2.5 ) = 74 .$] #solutionbox[ #math.equation(block: false, alt: "L equals 19.75")[$L = 19.75$] feet ] #strong[Stamps] Paula bought \$22.82 worth of 49-cent stamps and 21-cent stamps. The number of 21-cent stamps was eight less than the number of #linebreak() 49-cent stamps. Solve the equation #linebreak() #math.equation(block: false, alt: "0.49 s plus 0.21 ​ open parenthesis s minus 8 close parenthesis ​ ​ equals 22.82")[$0.49 s + 0.21 "​" ( s − 8 ) "​" "​" = 22.82$] for #emph[s], to find the number of 49-cent stamps Paula bought. ==== Writing Exercises Using your own words, list the steps in the general strategy for solving linear equations. #solutionbox[ Answers will vary. ] Explain why you should simplify both sides of an equation as much as possible before collecting the variable terms to one side and the constant terms to the other side. What is the first step you take when solving the equation #math.equation(block: false, alt: "3 minus 7 open parenthesis y minus 4 close parenthesis equals 38 ?")[$3 − 7 ( y − 4 ) = 38 ?$] Why is this your first step? #solutionbox[ Answers will vary. ] If an equation has several fractions, how does multiplying both sides by the LCD make it easier to solve? If an equation has fractions only on one side, why do you have to multiply both sides of the equation by the LCD? #solutionbox[ Answers will vary. ] For the equation #math.equation(block: false, alt: "0.35 x plus 2.1 equals 3.85 ,")[$0.35 x + 2.1 = 3.85 ,$] how do you clear the decimal? ==== Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. #figure(figph[This table has four columns and four rows. The first row is a header and it labels each column, “I can…”, “Confidently,” “With some help,” and “No-I don’t get it!” In row 2, the I can was solve linear equations using a general strategy. In row 3, the I can was classify equations. In row 4, the I can was solve equations with fraction or decimal coefficients.], alt: "This table has four columns and four rows. The first row is a header and it labels each column, “I can…”, “Confidently,” “With some help,” and “No-I don’t get it!” In row 2, the I can was solve linear equations using a general strategy. In row 3, the I can was classify equations. In row 4, the I can was solve equations with fraction or decimal coefficients.", caption: none) ⓑ If most of your checks were: …confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific. …with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help?Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved? …no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.