#set document(title: "2.4 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")) == 2.4#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: "− open parenthesis a minus 4 close parenthesis .")[$"−" ( a − 4 ) .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "minus a plus 4")[$− a + 4$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Multiply: #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"))[ Multiply: #math.equation(block: false, alt: "3 open parenthesis 7 y plus 9 close parenthesis")[$3 ( 7 y + 9 )$]. #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "21 y plus 27")[$21 y + 27$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Multiply: #math.equation(block: false, alt: "open parenthesis 2.5 close parenthesis open parenthesis 6.4 close parenthesis")[$( 2.5 ) ( 6.4 )$]. #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "16")[$16$] ] ] === Solve Equations Using the General Strategy Until now we have dealt with solving one specific form of a linear equation. It is time now to lay out one overall strategy that can be used to solve any linear equation. Some equations we solve will not require all these steps to solve, but many will. Beginning by simplifying each side of the equation makes the remaining steps easier. #examplebox("Example 1")[How to Solve Linear Equations Using the General Strategy][ Solve: #math.equation(block: false, alt: "−6 open parenthesis x plus 3 close parenthesis equals 24 .")[$−6 ( x + 3 ) = 24 .$] #solutionbox[ #figure(figph[This figure is a table that has three columns and five rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads: “Step 1. Simplify each side of the equation as much as possible.” The text in the second cell reads: “Use the Distributive Property. Notice that each side of the equation is simplified as much as possible.” The third cell contains the equation negative 6 times x plus 3, where x plus 3 is in parentheses, equals 24. Below this is the same equation with the negative 6 distributed across the parentheses: negative 6x minus 18 equals 24.], alt: "This figure is a table that has three columns and five rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads: “Step 1. Simplify each side of the equation as much as possible.” The text in the second cell reads: “Use the Distributive Property. Notice that each side of the equation is simplified as much as possible.” The third cell contains the equation negative 6 times x plus 3, where x plus 3 is in parentheses, equals 24. Below this is the same equation with the negative 6 distributed across the parentheses: negative 6x minus 18 equals 24.", caption: none) #figure(figph[In the second row of the table, the first cell says: “Step 2. Collect all variable terms on one side of the equation.” In the second cell, the instructions say: “Nothing to do—all x’s are on the left side. The third cell is blank.], alt: "In the second row of the table, the first cell says: “Step 2. Collect all variable terms on one side of the equation.” In the second cell, the instructions say: “Nothing to do—all x’s are on the left side. The third cell is blank.", caption: none) #figure(figph[In the third row of the table, the first cell says: “Step 3. Collect constant terms on the other side of the equation. In the second cell, the instructions say: “To get constants only on the right, add 18 to each side. Simplify.” The third cell contains the same equation with 18 added to both sides: negative 6x minus 18 plus 18 equals 24 plus 18. Below this is the equation negative 6x equals 42.], alt: "In the third row of the table, the first cell says: “Step 3. Collect constant terms on the other side of the equation. In the second cell, the instructions say: “To get constants only on the right, add 18 to each side. Simplify.” The third cell contains the same equation with 18 added to both sides: negative 6x minus 18 plus 18 equals 24 plus 18. Below this is the equation negative 6x equals 42.", caption: none) #figure(figph[In the fourth row of the table, the first cell says: “Step 4. Make the coefficient of the variable term equal to 1.” In the second cell, the instructions say: “Divide each side by negative 6. Simplify. The third cell contains the same equation divided by negative 6 on both sides: negative 6x over negative 6 equals 42 over negative 6, with “divided by negative 6” written in red on both sides. Below this is the answer to the equation: x equals negative 7.], alt: "In the fourth row of the table, the first cell says: “Step 4. Make the coefficient of the variable term equal to 1.” In the second cell, the instructions say: “Divide each side by negative 6. Simplify. The third cell contains the same equation divided by negative 6 on both sides: negative 6x over negative 6 equals 42 over negative 6, with “divided by negative 6” written in red on both sides. Below this is the answer to the equation: x equals negative 7.", caption: none) #figure(figph[In the fifth row of the table, the first cell says: “Step 5. Check the solution.” In the second cell, the instructions say: “Let x equal negative 7. Simplify. Multiply.” In the third cell, there is the instruction: “Check,” and to the right of this is the original equation again: negative 6 times x plus 3, with x plus 3 in parentheses, equal 24. Below this is the same equation with negative 7 substituted in for x: negative 6 times negative 7 plus 3, with negative 7 plus 3 in parentheses, might equal 24. Below this is the equation negative 6 times negative 4 might equal 24. Below this is the equation 24 equals 24, with a check mark next to it.], alt: "In the fifth row of the table, the first cell says: “Step 5. Check the solution.” In the second cell, the instructions say: “Let x equal negative 7. Simplify. Multiply.” In the third cell, there is the instruction: “Check,” and to the right of this is the original equation again: negative 6 times x plus 3, with x plus 3 in parentheses, equal 24. Below this is the same equation with negative 7 substituted in for x: negative 6 times negative 7 plus 3, with negative 7 plus 3 in parentheses, might equal 24. Below this is the equation negative 6 times negative 4 might equal 24. Below this is the equation 24 equals 24, with a check mark next to it.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "5 open parenthesis x plus 3 close parenthesis equals 35 .")[$5 ( x + 3 ) = 35 .$] #solutionbox[ #math.equation(block: true, alt: "x equals 4")[$x = 4$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "6 open parenthesis y minus 4 close parenthesis equals −18 .")[$6 ( y − 4 ) = −18 .$] #solutionbox[ #math.equation(block: true, alt: "y equals 1")[$y = 1$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[General strategy for solving linear equations.] + #strong[Simplify each side of the equation as much as possible.] #linebreak() Use the Distributive Property to remove any parentheses. #linebreak() Combine like terms. + #strong[Collect all the variable terms on one side of the equation.] #linebreak() Use the Addition or Subtraction Property of Equality. + #strong[Collect all the constant terms on the other side of the equation.] #linebreak() Use the Addition or Subtraction Property of Equality. + #strong[Make the coefficient of the variable term to equal to 1.] #linebreak() Use the Multiplication or Division Property of Equality. #linebreak() State the solution to the equation. + #strong[Check the solution.] Substitute the solution into the original equation to make sure the result is a true statement. ] #examplebox("Example 2")[][ Solve: #math.equation(block: false, alt: "− open parenthesis y plus 9 close parenthesis equals 8 .")[$"−" ( y + 9 ) = 8 .$] #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, [], [#figure(figph[A mathematical equation is displayed with the expression -(y + 9) = 8.], alt: "A mathematical equation is displayed with the expression -(y + 9) = 8.", caption: none)], [], [Simplify each side of the equation as much as possible by distributing.], [#figure(figph[A mathematical equation is displayed, reading '-y - 9 = 8' in a clear, dark font on a white background.], alt: "A mathematical equation is displayed, reading '-y - 9 = 8' in a clear, dark font on a white background.", caption: none)], [], [The only #math.equation(block: false, alt: "y")[$y$] term is on the left side, so all variable terms are on the left side of the equation.], [], [], [Add #math.equation(block: false, alt: "9")[$9$] to both sides to get all constant terms on the right side of the equation.], [#figure(figph[The equation -y - 9 + 9 = 8 + 9 is displayed, demonstrating how to add 9 to both sides to simplify the expression and move closer to solving for 'y'.], alt: "The equation -y - 9 + 9 = 8 + 9 is displayed, demonstrating how to add 9 to both sides to simplify the expression and move closer to solving for 'y'.", caption: none)], [], [Simplify.], [#figure(figph[The image shows a mathematical equation in black text on a white background, which states '-y = 17'.], alt: "The image shows a mathematical equation in black text on a white background, which states '-y = 17'.", caption: none)], [], [Rewrite #math.equation(block: false, alt: "minus y")[$− y$] as #math.equation(block: false, alt: "−1 y")[$−1 y$].], [#figure(figph[A mathematical equation is displayed with a white background. The equation reads '-1y = 17' in black text, demonstrating a simple linear equation where a negative coefficient multiplies the variable y, equaling 17.], alt: "A mathematical equation is displayed with a white background. The equation reads '-1y = 17' in black text, demonstrating a simple linear equation where a negative coefficient multiplies the variable y, equaling 17.", caption: none)], [], [Make the coefficient of the variable term to equal to #math.equation(block: false, alt: "1")[$1$] by dividing both sides by #math.equation(block: false, alt: "−1")[$−1$].], [#figure(figph[A mathematical equation showing both sides divided by -1: -1y / -1 = 17 / -1.], alt: "A mathematical equation showing both sides divided by -1: -1y / -1 = 17 / -1.", caption: none)], [], [Simplify.], [#figure(figph[The image displays the mathematical equation 'y = -17' in plain black text against a white background.], alt: "The image displays the mathematical equation 'y = -17' in plain black text against a white background.", caption: none)], [], [Check:], [#figure(figph[A mathematical equation is displayed, reading '-(y + 9) = 8' in black text on a white background. This equation involves a negative sign, parentheses, a variable 'y', and numbers.], alt: "A mathematical equation is displayed, reading '-(y + 9) = 8' in black text on a white background. This equation involves a negative sign, parentheses, a variable 'y', and numbers.", caption: none)], [], [Let #math.equation(block: false, alt: "y equals −17")[$y = −17$].], [#figure(figph[A mathematical equation asks if -(-17 + 9) equals 8, with '-17' in red. The correct evaluation shows that -(-8) = 8, so the statement is true.], alt: "A mathematical equation asks if -(-17 + 9) equals 8, with '-17' in red. The correct evaluation shows that -(-8) = 8, so the statement is true.", caption: none)], [], [], [#figure(figph[A math problem showing -(-8) = 8 with a question mark, asking to verify the equality.], alt: "A math problem showing -(-8) = 8 with a question mark, asking to verify the equality.", caption: none)], [], [], [#figure(figph[A mathematical expression displaying '8 = 8√'.], alt: "A mathematical expression displaying '8 = 8√'.", caption: none)], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "− open parenthesis y plus 8 close parenthesis equals −2 .")[$"−" ( y + 8 ) = −2 .$] #solutionbox[ #math.equation(block: true, alt: "y equals −6")[$y = −6$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "− open parenthesis z plus 4 close parenthesis equals −12 .")[$"−" ( z + 4 ) = −12 .$] #solutionbox[ #math.equation(block: true, alt: "z equals 8")[$z = 8$] ] ] #examplebox("Example 3")[][ Solve: #math.equation(block: false, alt: "5 open parenthesis a minus 3 close parenthesis plus 5 equals −10")[$5 ( a − 3 ) + 5 = −10$]. #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, [], [#figure(figph[A mathematical equation is displayed, showing 5(a - 3) + 5 = -10.], alt: "A mathematical equation is displayed, showing 5(a - 3) + 5 = -10.", caption: none)], [], [Simplify each side of the equation as much as possible.], [], [], [Distribute.], [#figure(figph[A mathematical equation is displayed, showing '5a - 15 + 5 = -10' on a white background.], alt: "A mathematical equation is displayed, showing '5a - 15 + 5 = -10' on a white background.", caption: none)], [], [Combine like terms.], [#figure(figph[A clear image of the algebraic equation: 5a - 10 = -10.], alt: "A clear image of the algebraic equation: 5a - 10 = -10.", caption: none)], [], [The only #math.equation(block: false, alt: "a")[$a$] term is on the left side, so all variable terms are on one side of the equation.], [], [], [Add #math.equation(block: false, alt: "10")[$10$] to both sides to get all constant terms on the other side of the equation.], [#figure(figph[A mathematical equation is displayed, showing 5a minus 10 plus 10 equals negative 10 plus 10. The addition of '10' on both sides of the equation is highlighted in red.], alt: "A mathematical equation is displayed, showing 5a minus 10 plus 10 equals negative 10 plus 10. The addition of '10' on both sides of the equation is highlighted in red.", caption: none)], [], [Simplify.], [#figure(figph[A mathematical equation on a white background showing '5a = 0'.], alt: "A mathematical equation on a white background showing '5a = 0'.", caption: none)], [], [Make the coefficient of the variable term to equal to #math.equation(block: false, alt: "1")[$1$] by dividing both sides by #math.equation(block: false, alt: "5")[$5$].], [#figure(figph[A mathematical equation showing 5a/5 = 0/5, where both sides of the equation are divided by 5 (in red font) to solve for 'a'.], alt: "A mathematical equation showing 5a/5 = 0/5, where both sides of the equation are divided by 5 (in red font) to solve for 'a'.", caption: none)], [], [Simplify.], [#figure(figph[A close-up of a mathematical equation, 'a = 0', presented in black text against a white background.], alt: "A close-up of a mathematical equation, 'a = 0', presented in black text against a white background.", caption: none)], [], [Check:], [#figure(figph[The image displays the algebraic equation 5(a - 3) + 5 = -10, which requires solving for the variable 'a'.], alt: "The image displays the algebraic equation 5(a - 3) + 5 = -10, which requires solving for the variable 'a'.", caption: none)], [], [Let #math.equation(block: false, alt: "a equals 0")[$a = 0$].], [#figure(figph[A mathematical equation is displayed, showing '5(0 - 3) + 5' on the left side, an equals sign with a question mark above it, and '-10' on the right side.], alt: "A mathematical equation is displayed, showing '5(0 - 3) + 5' on the left side, an equals sign with a question mark above it, and '-10' on the right side.", caption: none)], [], [], [#figure(figph[A mathematical equation is displayed, showing 5 multiplied by -3, plus 5, equals an unknown value indicated by a question mark, then equals -10. The full equation reads: 5(-3) + 5 =? -10.], alt: "A mathematical equation is displayed, showing 5 multiplied by -3, plus 5, equals an unknown value indicated by a question mark, then equals -10. The full equation reads: 5(-3) + 5 =? -10.", caption: none)], [], [], [#figure(figph[A mathematical equation: -15 + 5 = -10, with a question mark over the equals sign to inquire about its validity.], alt: "A mathematical equation: -15 + 5 = -10, with a question mark over the equals sign to inquire about its validity.", caption: none)], [], [], [#figure(figph[The image displays a simple mathematical equation, -10 = -10, followed by a checkmark, confirming its correctness on a white background.], alt: "The image displays a simple mathematical equation, -10 = -10, followed by a checkmark, confirming its correctness on a white background.", 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: "7 open parenthesis n minus 3 close parenthesis minus 8 equals −15")[$7 ( n − 3 ) − 8 = −15$]. #solutionbox[ #math.equation(block: true, alt: "n equals 2")[$n = 2$] ] ] #examplebox("Example 4")[][ Solve: #math.equation(block: false, alt: "the fraction 2 over 3 open parenthesis 6 m minus 3 close parenthesis equals 8 minus m")[$frac(2, 3) ( 6 m − 3 ) = 8 − m$]. #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, [], [#figure(figph[A mathematical equation is displayed: (2/3)(6m - 3) = 8 - m.], alt: "A mathematical equation is displayed: (2/3)(6m - 3) = 8 - m.", caption: none)], [], [Distribute.], [#figure(figph[A mathematical equation is displayed, showing '4m - 2 = 8 - m' in a plain font against a white background.], alt: "A mathematical equation is displayed, showing '4m - 2 = 8 - m' in a plain font against a white background.", caption: none)], [], [Add #math.equation(block: false, alt: "m")[$m$] to get the variables only to the left.], [#figure(figph[A mathematical equation is displayed with terms including the variable 'm', numbers, and arithmetic operators. The equation reads: '4m + m - 2 = 8 - m + m'.], alt: "A mathematical equation is displayed with terms including the variable 'm', numbers, and arithmetic operators. The equation reads: '4m + m - 2 = 8 - m + m'.", caption: none)], [], [Simplify.], [#figure(figph[A basic algebra equation is displayed, 5m - 2 = 8. This equation involves a variable 'm', showing a common mathematical problem where one needs to solve for the unknown.], alt: "A basic algebra equation is displayed, 5m - 2 = 8. This equation involves a variable 'm', showing a common mathematical problem where one needs to solve for the unknown.", caption: none)], [], [Add #math.equation(block: false, alt: "2")[$2$] to get constants only on the right.], [#figure(figph[An algebraic equation '5m - 2 + 2 = 8 + 2' is shown, with the number '2' highlighted in red on both sides, indicating an addition operation performed to balance the equation and solve for 'm'.], alt: "An algebraic equation '5m - 2 + 2 = 8 + 2' is shown, with the number '2' highlighted in red on both sides, indicating an addition operation performed to balance the equation and solve for 'm'.", caption: none)], [], [Simplify.], [#figure(figph[The image displays the algebraic equation '5m = 10' in dark gray text against a plain white background.], alt: "The image displays the algebraic equation '5m = 10' in dark gray text against a plain white background.", caption: none)], [], [Divide by #math.equation(block: false, alt: "5")[$5$].], [#figure(figph[A mathematical equation shows '5m over 5 equals 10 over 5', with the denominators '5' highlighted in red, indicating division on both sides of the equation.], alt: "A mathematical equation shows '5m over 5 equals 10 over 5', with the denominators '5' highlighted in red, indicating division on both sides of the equation.", caption: none)], [], [Simplify.], [#figure(figph[The image displays the equation 'm = 2' in black text against a plain white background, indicating a mathematical or scientific context.], alt: "The image displays the equation 'm = 2' in black text against a plain white background, indicating a mathematical or scientific context.", caption: none)], [], [Check:], [#figure(figph[A mathematical equation is displayed: 2/3(6m - 3) = 8 - m. It shows a linear equation with a variable 'm' to be solved, involving a fraction, parentheses, and distribution.], alt: "A mathematical equation is displayed: 2/3(6m - 3) = 8 - m. It shows a linear equation with a variable 'm' to be solved, involving a fraction, parentheses, and distribution.", caption: none)], [], [Let #math.equation(block: false, alt: "m equals 2")[$m = 2$].], [#figure(figph[A math problem is presented, asking to verify if the expression (2/3)(6 \* 2 - 3) equals 8 - 2. Numbers '2' on both sides are highlighted in red, with a question mark over the equality sign.], alt: "A math problem is presented, asking to verify if the expression (2/3)(6 * 2 - 3) equals 8 - 2. Numbers '2' on both sides are highlighted in red, with a question mark over the equality sign.", caption: none)], [], [], [#figure(figph[A mathematical equation asks if two-thirds multiplied by the difference of twelve and three is equal to six. The expression shown is 2/3(12-3) ?= 6, with a question mark over the equals sign.], alt: "A mathematical equation asks if two-thirds multiplied by the difference of twelve and three is equal to six. The expression shown is 2/3(12-3) ?= 6, with a question mark over the equals sign.", caption: none)], [], [], [#figure(figph[A math problem showing the expression (2/3)(9) with an equals sign followed by a question mark and the number 6, asking if two-thirds of nine is equal to six.], alt: "A math problem showing the expression (2/3)(9) with an equals sign followed by a question mark and the number 6, asking if two-thirds of nine is equal to six.", caption: none)], [], [], [#figure(figph[The image shows the mathematical equation '6 = 6' followed by a checkmark, indicating its correctness.], alt: "The image shows the mathematical equation '6 = 6' followed by a checkmark, indicating its correctness.", 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")[$x = 4$] ] ] #examplebox("Example 5")[][ Solve: #math.equation(block: false, alt: "8 minus 2 open parenthesis 3 y plus 5 close parenthesis equals 0")[$8 − 2 ( 3 y + 5 ) = 0$]. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[An algebraic equation is shown on a white background. The equation reads as follows: 8 - 2(3y + 5) = 0.], alt: "An algebraic equation is shown on a white background. The equation reads as follows: 8 - 2(3y + 5) = 0.", caption: none)]), [Simplify—use the Distributive Property.], [#figure(figph[A mathematical equation is displayed, reading '8 - 6y - 10 = 0'.], alt: "A mathematical equation is displayed, reading '8 - 6y - 10 = 0'.", caption: none)], [Combine like terms.], [#figure(figph[The image displays the linear equation -6y - 2 = 0, presented in a clear, standard mathematical format against a plain white background.], alt: "The image displays the linear equation -6y - 2 = 0, presented in a clear, standard mathematical format against a plain white background.", caption: none)], [Add #math.equation(block: false, alt: "2")[$2$] to both sides to collect constants on the right.], [#figure(figph[A mathematical equation shows '-6y - 2 + 2 = 0 + 2' with the second '+ 2' on both sides highlighted in red.], alt: "A mathematical equation shows '-6y - 2 + 2 = 0 + 2' with the second '+ 2' on both sides highlighted in red.", caption: none)], [Simplify.], [#figure(figph[A mathematical equation is displayed, showing -6y = 2.], alt: "A mathematical equation is displayed, showing -6y = 2.", caption: none)], [Divide both sides by #math.equation(block: false, alt: "−6")[$−6$].], [#figure(figph[The equation -6y / -6 = 2 / -6, showing both sides divided by -6 to solve for 'y'.], alt: "The equation -6y / -6 = 2 / -6, showing both sides divided by -6 to solve for 'y'.", caption: none)], [Simplify.], [#figure(figph[The image displays a mathematical equation: y equals negative one-third (y = -1/3), presented in a clear, digital font on a white background, suggesting a constant horizontal line.], alt: "The image displays a mathematical equation: y equals negative one-third (y = -1/3), presented in a clear, digital font on a white background, suggesting a constant horizontal line.", caption: none)], [Check: Let #math.equation(block: false, alt: "y equals minus the fraction 1 over 3 .")[$y = − frac(1, 3) .$] #linebreak() #figure(figph[Verification steps for the equation 8 - 2(3y + 5) = 0. The process shows that substituting y = -1/3 results in 0 = 0, confirming it as the correct solution with a checkmark.], alt: "Verification steps for the equation 8 - 2(3y + 5) = 0. The process shows that substituting y = -1/3 results in 0 = 0, confirming it as the correct solution with a checkmark.", caption: none)], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "12 minus 3 open parenthesis 4 j plus 3 close parenthesis equals −17")[$12 − 3 ( 4 j + 3 ) = −17$]. #solutionbox[ #math.equation(block: true, alt: "j equals the fraction 5 over 3")[$j = frac(5, 3)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "−6 minus 8 open parenthesis k minus 2 close parenthesis equals −10")[$−6 − 8 ( k − 2 ) = −10$]. #solutionbox[ #math.equation(block: true, alt: "k equals the fraction 5 over 2")[$k = frac(5, 2)$] ] ] #examplebox("Example 6")[][ 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: 3, align: left, inset: 6pt, [], [#figure(figph[A mathematical equation is displayed, reading 4(x - 1) - 2 = 5(2x + 3) + 6.], alt: "A mathematical equation is displayed, reading 4(x - 1) - 2 = 5(2x + 3) + 6.", caption: none)], [], [Distribute.], [#figure(figph[A mathematical equation is displayed with terms including 4x, -4, -2, equals sign, 10x, +15, and +6, for the equation 4x - 4 - 2 = 10x + 15 + 6.], alt: "A mathematical equation is displayed with terms including 4x, -4, -2, equals sign, 10x, +15, and +6, for the equation 4x - 4 - 2 = 10x + 15 + 6.", caption: none)], [], [Combine like terms.], [#figure(figph[A mathematical equation is displayed: 4x - 6 = 10x + 21. The equation shows a linear equation with 'x' as the variable on both sides.], alt: "A mathematical equation is displayed: 4x - 6 = 10x + 21. The equation shows a linear equation with 'x' as the variable on both sides.", caption: none)], [], [Subtract #math.equation(block: false, alt: "4 x")[$4 x$] to get the variables only on the right side since #math.equation(block: false, alt: "10 greater than 4")[$10 > 4$].], [#figure(figph[A mathematical equation showing 4x minus 4x minus 6 equals 10x minus 4x plus 21, with the '4x' terms highlighted in red to indicate they cancel out or are being subtracted.], alt: "A mathematical equation showing 4x minus 4x minus 6 equals 10x minus 4x plus 21, with the '4x' terms highlighted in red to indicate they cancel out or are being subtracted.", caption: none)], [], [Simplify.], [#figure(figph[A mathematical equation is displayed on a white background, reading '-6 = 6x + 21'.], alt: "A mathematical equation is displayed on a white background, reading '-6 = 6x + 21'.", caption: none)], [], [Subtract #math.equation(block: false, alt: "21")[$21$] to get the constants on left.], [#figure(figph[Simplifying an equation: subtracting 21 from both sides of -6 - 21 = 6x + 21 - 21 to isolate the variable term.], alt: "Simplifying an equation: subtracting 21 from both sides of -6 - 21 = 6x + 21 - 21 to isolate the variable term.", caption: none)], [], [Simplify.], [#figure(figph[A mathematical equation is displayed with the expression -27 = 6x, featuring white text against a plain white background.], alt: "A mathematical equation is displayed with the expression -27 = 6x, featuring white text against a plain white background.", caption: none)], [], [Divide by 6.], [#figure(figph[A mathematical equation displaying two fractions set equal to each other: -27/6 = 6x/6. The denominators in both fractions are 6.], alt: "A mathematical equation displaying two fractions set equal to each other: -27/6 = 6x/6. The denominators in both fractions are 6.", caption: none)], [], [Simplify.], [#figure(figph[A mathematical equation is displayed against a white background, reading '-9/2 = x' in black text.], alt: "A mathematical equation is displayed against a white background, reading '-9/2 = x' in black text.", caption: none)], [], [Check:], [#figure(figph[An algebraic equation is shown: 4(x - 1) - 2 = 5(2x + 3) + 6.], alt: "An algebraic equation is shown: 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 is displayed, asking whether 4(-9/2 - 1) - 2 is equal to 5\[2(-9/2) + 3\] + 6. The -9/2 is highlighted in red, indicating a common term or emphasis.], alt: "A mathematical equation is displayed, asking whether 4(-9/2 - 1) - 2 is equal to 5[2(-9/2) + 3] + 6. The -9/2 is highlighted in red, indicating a common term or emphasis.", caption: none)], [], [], [#figure(figph[A mathematical equation is displayed, asking to check if 4 multiplied by negative eleven-halves, minus 2, is equal to 5 multiplied by the sum of negative 9 and 3, plus 6.], alt: "A mathematical equation is displayed, asking to check if 4 multiplied by negative eleven-halves, minus 2, is equal to 5 multiplied by the sum of negative 9 and 3, plus 6.", caption: none)], [], [], [#figure(figph[A mathematical equation is displayed: -22 - 2 ?= 5(-6) + 6. Both sides of the equality, when calculated, result in -24, confirming that the equation is true.], alt: "A mathematical equation is displayed: -22 - 2 ?= 5(-6) + 6. Both sides of the equality, when calculated, result in -24, confirming that the equation is true.", caption: none)], [], [], [#figure(figph[A mathematical problem asks to verify if -24 equals -30 plus 6, with a question mark above the equal sign. The expression -30 + 6 simplifies to -24, making the equation true.], alt: "A mathematical problem asks to verify if -24 equals -30 plus 6, with a question mark above the equal sign. The expression -30 + 6 simplifies to -24, making the equation true.", caption: none)], [], [], [#figure(figph[The equation -24 = -24 is displayed with a checkmark, indicating the mathematical statement is correct.], alt: "The equation -24 = -24 is displayed with a checkmark, indicating the mathematical statement 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 7")[][ 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 multi-step linear equation: 10\[3 - 8(2s - 5)\] = 15(40 - 5s), requiring algebraic manipulation to find 's'.], alt: "A multi-step linear equation: 10[3 - 8(2s - 5)] = 15(40 - 5s), requiring algebraic manipulation to find 's'.", caption: none)], [], [Simplify from the innermost parentheses first.], [#figure(figph[A mathematical equation is displayed: 10\[3 - 16s + 40\] = 15(40 - 5s).], alt: "A mathematical equation is displayed: 10[3 - 16s + 40] = 15(40 - 5s).", caption: none)], [], [Combine like terms in the brackets.], [#figure(figph[A mathematical equation is displayed on a white background: 10\[43 - 16s\] = 15(40 - 5s).], alt: "A mathematical equation is displayed on a white background: 10[43 - 16s] = 15(40 - 5s).", caption: none)], [], [Distribute.], [#figure(figph[A mathematical equation is displayed: 430 - 160s = 600 - 75s.], alt: "A mathematical equation is displayed: 430 - 160s = 600 - 75s.", caption: none)], [], [Add #math.equation(block: false, alt: "160 s")[$160 s$] to get the s’s to the right.], [#figure(figph[A mathematical equation reads 430 - 160s + 160s = 600 - 75s + 160s, demonstrating the addition of 160s to both sides to isolate the 's' variable on one side of the equality.], alt: "A mathematical equation reads 430 - 160s + 160s = 600 - 75s + 160s, demonstrating the addition of 160s to both sides to isolate the 's' variable on one side of the equality.", caption: none)], [], [Simplify.], [#figure(figph[A mathematical equation is displayed on a white background, reading 430 = 600 + 85s.], alt: "A mathematical equation is displayed on a white background, reading 430 = 600 + 85s.", caption: none)], [], [Subtract 600 to get the constants to the left.], [#figure(figph[A mathematical equation shows '430 - 600 = 600 + 85s - 600', with the number '600' highlighted in red on both sides of the equals sign, indicating a step in solving for 's'.], alt: "A mathematical equation shows '430 - 600 = 600 + 85s - 600', with the number '600' highlighted in red on both sides of the equals sign, indicating a step in solving for 's'.", 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.], [#figure(figph[The equation -170/85 = 85s/85, demonstrating the step of dividing both sides by 85 to solve for 's'.], alt: "The equation -170/85 = 85s/85, demonstrating the step of dividing both sides by 85 to solve for 's'.", caption: none)], [], [Simplify.], [#figure(figph[A mathematical expression showing '-2=S' in a black font on a plain white background.], alt: "A mathematical expression showing '-2=S' in a black font on a plain white background.", caption: none)], [], [Check:], [#figure(figph[A mathematical equation is displayed, reading '10\[3 - 8(2s - 5)\] = 15(40 - 5s)'.], alt: "A mathematical equation is displayed, reading '10[3 - 8(2s - 5)] = 15(40 - 5s)'.", caption: none)], [], [Substitute #math.equation(block: false, alt: "s equals −2")[$s = −2$].], [#figure(figph[A mathematical equation is displayed: 10\[3 - 8(2(-2) - 5)\] ?= 15(40 - 5(-2)). Numbers with negative signs are highlighted in red.], alt: "A mathematical equation is displayed: 10[3 - 8(2(-2) - 5)] ?= 15(40 - 5(-2)). Numbers with negative signs are highlighted in red.", caption: none)], [], [], [#figure(figph[A math problem showing 10\[3 - 8(-4 - 5)\] with a question mark over the equals sign and 15(40 + 10), challenging the viewer to determine if the two sides are equal.], alt: "A math problem showing 10[3 - 8(-4 - 5)] with a question mark over the equals sign and 15(40 + 10), challenging the viewer to determine if the two sides are equal.", caption: none)], [], [], [#figure(figph[A mathematical equation: 10\[3 - 8(-9)\] =? 15(50). This problem tests the order of operations, requiring calculation of both sides to determine if they are equal, indicated by the question mark above the equal sign.], alt: "A mathematical equation: 10[3 - 8(-9)] =? 15(50). This problem tests the order of operations, requiring calculation of both sides to determine if they are equal, indicated by the question mark above the equal sign.", caption: none)], [], [], [#figure(figph[A mathematical equation asks if 10 multiplied by the sum of 3 and 72 equals 750. Evaluating 10\[3 + 72\] gives 10\[75\], which is indeed 750. The equation is true.], alt: "A mathematical equation asks if 10 multiplied by the sum of 3 and 72 equals 750. Evaluating 10[3 + 72] gives 10[75], which is indeed 750. The equation is true.", caption: none)], [], [], [#figure(figph[A mathematical expression '10\[75\] =? 750' is displayed on a white background, posing a question about whether 10 multiplied by 75 equals 750.], alt: "A mathematical expression '10[75] =? 750' is displayed on a white background, posing a question about whether 10 multiplied by 75 equals 750.", caption: none)], [], [], [#figure(figph[The number 750 equals 750, confirmed by a checkmark on a white background, representing a verified mathematical statement.], alt: "The number 750 equals 750, confirmed by a checkmark on a white background, representing a verified mathematical statement.", 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")[$z = 0$] ] ] #examplebox("Example 8")[][ Solve: #math.equation(block: false, alt: "0.36 open parenthesis 100 n plus 5 close parenthesis equals 0.6 open parenthesis 30 n plus 15 close parenthesis")[$0.36 ( 100 n + 5 ) = 0.6 ( 30 n + 15 )$]. #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, [], [#figure(figph[A mathematical equation shows 0.36 multiplied by the quantity (100n plus 5), set equal to 0.6 multiplied by the quantity (30n plus 15).], alt: "A mathematical equation shows 0.36 multiplied by the quantity (100n plus 5), set equal to 0.6 multiplied by the quantity (30n plus 15).", caption: none)], [], [Distribute.], [#figure(figph[An algebraic equation is shown: 36n + 1.8 = 18n + 9.], alt: "An algebraic equation is shown: 36n + 1.8 = 18n + 9.", caption: none)], [], [Subtract #math.equation(block: false, alt: "18 n")[$18 n$] to get the variables to the left.], [#figure(figph[A mathematical equation displayed on a white background: 36n - 18n + 1.8 = 18n - 18n + 9, with the '18n' terms highlighted in red.], alt: "A mathematical equation displayed on a white background: 36n - 18n + 1.8 = 18n - 18n + 9, with the '18n' terms highlighted in red.", caption: none)], [], [Simplify.], [#figure(figph[A mathematical equation is displayed on a white background, which reads], alt: "A mathematical equation is displayed on a white background, which reads", caption: none)], [], [Subtract #math.equation(block: false, alt: "1.8")[$1.8$] to get the constants to the right.], [#figure(figph[A mathematical equation is displayed against a white background: '18n + 1.8 - 1.8 = 9 - 1.8'. The numbers '- 1.8' are highlighted in red on both sides of the equation.], alt: "A mathematical equation is displayed against a white background: '18n + 1.8 - 1.8 = 9 - 1.8'. The numbers '- 1.8' are highlighted in red on both sides of the equation.", caption: none)], [], [Simplify.], [#figure(figph[A mathematical equation is displayed on a white background, which reads '18n = 7.2' in black characters. This equation represents a basic algebraic problem.], alt: "A mathematical equation is displayed on a white background, which reads '18n = 7.2' in black characters. This equation represents a basic algebraic problem.", caption: none)], [], [Divide.], [#figure(figph[A step in solving an algebraic equation shows both sides of the equation being divided by 18: (18n)/18 = 7.2/18, with the denominators highlighted in red.], alt: "A step in solving an algebraic equation shows both sides of the equation being divided by 18: (18n)/18 = 7.2/18, with the denominators highlighted in red.", caption: none)], [], [Simplify.], [#figure(figph[The image displays the equation 'n = 0.4' centered on a plain white background, rendered in a standard, clear gray typeface.], alt: "The image displays the equation 'n = 0.4' centered on a plain white background, rendered in a standard, clear gray typeface.", caption: none)], [], [Check:], [#figure(figph[A mathematical equation is displayed on a white background: 0.36(100n + 5) = 0.6(30n + 15).], alt: "A mathematical equation is displayed on a white background: 0.36(100n + 5) = 0.6(30n + 15).", caption: none)], [], [Let #math.equation(block: false, alt: "n equals 0.4")[$n = 0.4$].], [#figure(figph[A mathematical equation is displayed, questioning if 0.36 multiplied by (100 times 0.4 plus 5) equals 0.6 multiplied by (30 times 0.4 plus 15).], alt: "A mathematical equation is displayed, questioning if 0.36 multiplied by (100 times 0.4 plus 5) equals 0.6 multiplied by (30 times 0.4 plus 15).", caption: none)], [], [], [#figure(figph[A mathematical equation shows '0.36(40 + 5) ?= 0.6(12 + 15)', asking if the expressions on both sides are equal. The question mark above the equals sign indicates that the equality needs to be verified.], alt: "A mathematical equation shows '0.36(40 + 5) ?= 0.6(12 + 15)', asking if the expressions on both sides are equal. The question mark above the equals sign indicates that the equality needs to be verified.", caption: none)], [], [], [#figure(figph[A mathematical equation is displayed, asking whether 0.36 multiplied by 45 is equal to 0.6 multiplied by 27, denoted by a question mark over the equals sign.], alt: "A mathematical equation is displayed, asking whether 0.36 multiplied by 45 is equal to 0.6 multiplied by 27, denoted by a question mark over the equals sign.", caption: none)], [], [], [#figure(figph[The mathematical expression '16.2 = 16.2√' is shown on a white background, appearing as an incomplete or incorrect equation due to the trailing square root symbol.], alt: "The mathematical expression '16.2 = 16.2√' is shown on a white background, appearing as an incomplete or incorrect equation due to the trailing square root symbol.", caption: none)], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "0.55 open parenthesis 100 n plus 8 close parenthesis equals 0.6 open parenthesis 85 n plus 14 close parenthesis")[$0.55 ( 100 n + 8 ) = 0.6 ( 85 n + 14 )$]. #solutionbox[ #math.equation(block: true, alt: "n equals 1")[$n = 1$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "0.15 open parenthesis 40 m minus 120 close parenthesis equals 0.5 open parenthesis 60 m plus 12 close parenthesis")[$0.15 ( 40 m − 120 ) = 0.5 ( 60 m + 12 )$]. #solutionbox[ #math.equation(block: true, alt: "m equals −1")[$m = −1$] ] ] === Classify Equations Consider the equation we solved at the start of the last section, #math.equation(block: false, alt: "7 x plus 8 equals −13")[$7 x + 8 = −13$]. The solution we found was #math.equation(block: false, alt: "x equals −3")[$x = −3$]. This means 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")[$−3$]. We showed this when we checked the solution #math.equation(block: false, alt: "x equals −3")[$x = −3$] and evaluated #math.equation(block: false, alt: "7 x plus 8 equals −13")[$7 x + 8 = −13$] for #math.equation(block: false, alt: "x equals −3")[$x = −3$]. #figure(figph[This figure shows why we can say the equation 7x plus 8 equals negative 13 is true when the variable x is replaced with the value negative 3. The first line shows the equation with negative 3 substituted in for x: 7 times negative 3 plus 8 might equal negative 13. Below this is the equation negative 21 plus 8 might equal negative 13. Below this is the equation negative 13 equals negative 13, with a check mark next to it.], alt: "This figure shows why we can say the equation 7x plus 8 equals negative 13 is true when the variable x is replaced with the value negative 3. The first line shows the equation with negative 3 substituted in for x: 7 times negative 3 plus 8 might equal negative 13. Below this is the equation negative 21 plus 8 might equal negative 13. Below this is the equation negative 13 equals negative 13, with a check mark next to it.", caption: none) If we evaluate #math.equation(block: false, alt: "7 x plus 8")[$7 x + 8$] for a different value of #emph[x], the left side will not be #math.equation(block: false, alt: "−13")[$−13$]. 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. Whether or not the equation #math.equation(block: false, alt: "7 x plus 8 equals −13")[$7 x + 8 = −13$] is true depends on the value of the variable. Equations like this are called conditional equations. 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: "2 y plus 6 equals 2 open parenthesis y plus 3 close parenthesis")[$2 y + 6 = 2 ( y + 3 )$]. Do you recognize that the left side and the right side are equivalent? Let’s see what happens when we solve for #emph[y]. #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The image displays the algebraic equation 2y + 6 = 2(y + 3).], alt: "The image displays the algebraic equation 2y + 6 = 2(y + 3).", caption: none)]), [Distribute.], [#figure(figph[The equation 2y + 6 = 2y + 6, an identity that is true for all values of y.], alt: "The equation 2y + 6 = 2y + 6, an identity that is true for all values of y.", caption: none)], [Subtract #math.equation(block: false, alt: "2 y")[$2 y$] to get the #math.equation(block: false, alt: "y")[$y$]’s to one side.], [#figure(figph[An algebraic identity: 2y - 2y + 6 = 2y - 2y + 6. The -2y on both sides is highlighted in red, showing how variables cancel out to leave 6 = 6.], alt: "An algebraic identity: 2y - 2y + 6 = 2y - 2y + 6. The -2y on both sides is highlighted in red, showing how variables cancel out to leave 6 = 6.", caption: none)], [Simplify—the #math.equation(block: false, alt: "y")[$y$]’s are gone!], [#figure(figph[The number six is equal to the number six, shown as '6 = 6' in black text on a white background.], alt: "The number six is equal to the number six, shown as '6 = 6' in black text on a white background.", caption: none)], )) But #math.equation(block: false, alt: "6 equals 6")[$6 = 6$] is true. This means that the equation #math.equation(block: false, alt: "2 y plus 6 equals 2 open parenthesis y plus 3 close parenthesis")[$2 y + 6 = 2 ( y + 3 )$] 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 like this 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[#strong[identity]]. The solution of an identity is all real numbers. ] What happens when we solve the equation #math.equation(block: false, alt: "5 z equals 5 z minus 1")[$5 z = 5 z − 1$]? #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The equation 5z = 5z - 1 is displayed, a mathematical contradiction implying that there is no value of 'z' for which this statement is true. It simplifies to 0 = -1, indicating no solution.], alt: "The equation 5z = 5z - 1 is displayed, a mathematical contradiction implying that there is no value of 'z' for which this statement is true. It simplifies to 0 = -1, indicating no solution.", caption: none)]), [Subtract #math.equation(block: false, alt: "5 z")[$5 z$] to get the constant alone on the right.], [#figure(figph[A mathematical equation reads '5z - 5z = 5z - 5z - 1', which simplifies to 0 = -1, representing an impossible or false statement in algebra.], alt: "A mathematical equation reads '5z - 5z = 5z - 5z - 1', which simplifies to 0 = -1, representing an impossible or false statement in algebra.", caption: none)], [Simplify—the #math.equation(block: false, alt: "z")[$z$]’s are gone!], [#figure(figph[The mathematical expression 0 eq -1 is displayed in a simple, clear font against a white background.], alt: "The mathematical expression 0 eq -1 is displayed in a simple, clear font against a white background.", caption: none)], )) But #math.equation(block: false, alt: "0 not equal to − 1")[$0 ≠ "−" 1$]. Solving the equation #math.equation(block: false, alt: "5 z equals 5 z minus 1")[$5 z = 5 z − 1$] led to the false statement #math.equation(block: false, alt: "0 equals −1")[$0 = −1$]. The equation #math.equation(block: false, alt: "5 z equals 5 z minus 1")[$5 z = 5 z − 1$] will not be true for any value of 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. ] #examplebox("Example 9")[][ Classify the equation as a conditional equation, an identity, or a contradiction. Then state the solution. #math.equation(block: true, 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 on a white background. The equation is 6(2n - 1) + 3 = 2n - 8 + 5(2n + 1).], alt: "A mathematical equation is displayed on a white background. The equation is 6(2n - 1) + 3 = 2n - 8 + 5(2n + 1).", caption: none)]), [Distribute.], [#figure(figph[The algebraic equation 12n - 6 + 3 = 2n - 8 + 10n + 5 is an identity, meaning it is true for all values of 'n'.], alt: "The algebraic equation 12n - 6 + 3 = 2n - 8 + 10n + 5 is an identity, meaning it is true for all values of 'n'.", caption: none)], [Combine like terms.], [#figure(figph[The image displays the equation 12n - 3 = 12n - 3 in the center of a plain white background. The text is rendered in a dark grey, sans-serif font, clearly showing an identical expression on both sides of the equals sign.], alt: "The image displays the equation 12n - 3 = 12n - 3 in the center of a plain white background. The text is rendered in a dark grey, sans-serif font, clearly showing an identical expression on both sides of the equals sign.", caption: none)], [Subtract #math.equation(block: false, alt: "12 n")[$12 n$] to get the #math.equation(block: false, alt: "n")[$n$]’s to one side.], [#figure(figph[A mathematical equation displays '12n - 12n - 3 = 12n - 12n - 3', with the 12n terms in black, the -12n terms in red, and the -3 terms in black, illustrating an algebraic identity.], alt: "A mathematical equation displays '12n - 12n - 3 = 12n - 12n - 3', with the 12n terms in black, the -12n terms in red, and the -3 terms in black, illustrating an algebraic identity.", caption: none)], [Simplify.], [#figure(figph[A mathematical equation, '-3 = -3', is displayed in black text against a plain white background.], alt: "A mathematical equation, '-3 = -3', is displayed in black text against a plain white background.", caption: none)], [This is a true statement.], [The equation is an identity. #linebreak() 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: true, 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: true, 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 10")[][ Classify as a conditional equation, an identity, or a contradiction. Then state the solution. #math.equation(block: true, alt: "10 plus 4 open parenthesis p minus 5 close parenthesis equals 0")[$10 + 4 ( p − 5 ) = 0$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical equation is displayed on a white background: 10 + 4(p - 5) = 0.], alt: "A mathematical equation is displayed on a white background: 10 + 4(p - 5) = 0.", caption: none)]), [Distribute.], [#figure(figph[An algebraic equation is shown, displaying '10 + 4p - 20 = 0'.], alt: "An algebraic equation is shown, displaying '10 + 4p - 20 = 0'.", caption: none)], [Combine like terms.], [#figure(figph[A mathematical equation is displayed, showing '4p - 10 = 0' in a simple, clear font on a white background.], alt: "A mathematical equation is displayed, showing '4p - 10 = 0' in a simple, clear font on a white background.", caption: none)], [Add #math.equation(block: false, alt: "10")[$10$] to both sides.], [#figure(figph[A mathematical equation shows '4p - 10 + 10 = 0 + 10,' illustrating the process of adding 10 to both sides of an equation, with the added '+ 10' highlighted in red.], alt: "A mathematical equation shows '4p - 10 + 10 = 0 + 10,' illustrating the process of adding 10 to both sides of an equation, with the added '+ 10' highlighted in red.", caption: none)], [Simplify.], [#figure(figph[The image displays the algebraic equation 4p = 10, representing a linear equation where the variable 'p' is multiplied by 4 and set equal to 10.], alt: "The image displays the algebraic equation 4p = 10, representing a linear equation where the variable 'p' is multiplied by 4 and set equal to 10.", caption: none)], [Divide.], [#figure(figph[A mathematical equation is shown where both sides are divided by 4: '4p' divided by '4' equals '10' divided by '4'. The '4' in the denominator on both sides is highlighted in red.], alt: "A mathematical equation is shown where both sides are divided by 4: '4p' divided by '4' equals '10' divided by '4'. The '4' in the denominator on both sides is highlighted in red.", caption: none)], [Simplify.], [#figure(figph[The mathematical equation p = 5/2 is displayed against a white background.], alt: "The mathematical equation p = 5/2 is displayed against a white background.", caption: none)], [The equation is true when #math.equation(block: false, alt: "p equals the fraction 5 over 2")[$p = frac(5, 2)$].], [This is a conditional equation. #linebreak() The solution is #math.equation(block: false, alt: "p equals the fraction 5 over 2 .")[$p = frac(5, 2) .$]], )) ] ] #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 plus 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 11")[][ Classify the equation as a conditional equation, an identity, or a contradiction. Then state the solution. #math.equation(block: true, 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[A mathematical equation is displayed, which reads '5m + 3(9 + 3m) = 2(7m - 11)'. The equation is written in black text on a white background.], alt: "A mathematical equation is displayed, which reads '5m + 3(9 + 3m) = 2(7m - 11)'. The equation is written in black text on a white background.", caption: none)]), [Distribute.], [#figure(figph[An algebraic equation, 5m + 27 + 9m = 14m - 22, displayed on a white background.], alt: "An algebraic equation, 5m + 27 + 9m = 14m - 22, displayed on a white background.", caption: none)], [Combine like terms.], [#figure(figph[A mathematical equation is displayed on a white background: 14m + 27 = 14m - 22. This equation has no solution as simplifying it leads to 27 = -22, which is false.], alt: "A mathematical equation is displayed on a white background: 14m + 27 = 14m - 22. This equation has no solution as simplifying it leads to 27 = -22, which is false.", caption: none)], [Subtract #math.equation(block: false, alt: "14 m")[$14 m$] from both sides.], [#figure(figph[A mathematical equation is shown: 14m + 27 - 14m = 14m - 22 - 14m. The '-14m' terms on both sides of the equality are highlighted in red.], alt: "A mathematical equation is shown: 14m + 27 - 14m = 14m - 22 - 14m. The '-14m' terms on both sides of the equality are highlighted in red.", caption: none)], [Simplify.], [#figure(figph[The mathematical expression '27 = -22' is shown with a strike-through on the equals sign, indicating that 27 is not equal to -22.], alt: "The mathematical expression '27 = -22' is shown with a strike-through on the equals sign, indicating that 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. #linebreak() 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: true, 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: true, 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 ] ] #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]], [#strong[True] for any value of the variable], [All real numbers], [#strong[Contradiction]], [#strong[False] for all values of the variable], [No solution], )) === Key Concepts - #strong[General Strategy for Solving Linear Equations] + 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 to 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. ==== Practice Makes Perfect #strong[Solve Equations Using the General Strategy for Solving Linear Equations] In the following exercises, solve each linear equation. #math.equation(block: true, alt: "15 open parenthesis y minus 9 close parenthesis equals −60")[$15 ( y − 9 ) = −60$] #math.equation(block: true, alt: "21 open parenthesis y minus 5 close parenthesis equals −42")[$21 ( y − 5 ) = −42$] #solutionbox[ #math.equation(block: true, alt: "y equals 3")[$y = 3$] ] #math.equation(block: true, alt: "−9 open parenthesis 2 n plus 1 close parenthesis equals 36")[$−9 ( 2 n + 1 ) = 36$] #math.equation(block: true, alt: "−16 open parenthesis 3 n plus 4 close parenthesis equals 32")[$−16 ( 3 n + 4 ) = 32$] #solutionbox[ #math.equation(block: true, alt: "n equals −2")[$n = −2$] ] #math.equation(block: true, alt: "8 open parenthesis 22 plus 11 r close parenthesis equals 0")[$8 ( 22 + 11 r ) = 0$] #math.equation(block: true, alt: "5 open parenthesis 8 plus 6 p close parenthesis equals 0")[$5 ( 8 + 6 p ) = 0$] #solutionbox[ #math.equation(block: true, alt: "p equals minus the fraction 4 over 3")[$p = − frac(4, 3)$] ] #math.equation(block: true, alt: "− open parenthesis w minus 12 close parenthesis equals 30")[$"−" ( w − 12 ) = 30$] #math.equation(block: true, alt: "− open parenthesis t minus 19 close parenthesis equals 28")[$"−" ( t − 19 ) = 28$] #solutionbox[ #math.equation(block: true, alt: "t equals −9")[$t = −9$] ] #math.equation(block: true, alt: "9 open parenthesis 6 a plus 8 close parenthesis plus 9 equals 81")[$9 ( 6 a + 8 ) + 9 = 81$] #math.equation(block: true, alt: "8 open parenthesis 9 b minus 4 close parenthesis minus 12 equals 100")[$8 ( 9 b − 4 ) − 12 = 100$] #solutionbox[ #math.equation(block: true, alt: "b equals 2")[$b = 2$] ] #math.equation(block: true, alt: "32 plus 3 open parenthesis z plus 4 close parenthesis equals 41")[$32 + 3 ( z + 4 ) = 41$] #math.equation(block: true, alt: "21 plus 2 open parenthesis m minus 4 close parenthesis equals 25")[$21 + 2 ( m − 4 ) = 25$] #solutionbox[ #math.equation(block: true, alt: "m equals 6")[$m = 6$] ] #math.equation(block: true, alt: "51 plus 5 open parenthesis 4 minus q close parenthesis equals 56")[$51 + 5 ( 4 − q ) = 56$] #math.equation(block: true, alt: "−6 plus 6 open parenthesis 5 minus k close parenthesis equals 15")[$−6 + 6 ( 5 − k ) = 15$] #solutionbox[ #math.equation(block: true, alt: "k equals the fraction 3 over 2")[$k = frac(3, 2)$] ] #math.equation(block: true, alt: "2 open parenthesis 9 s minus 6 close parenthesis minus 62 equals 16")[$2 ( 9 s − 6 ) − 62 = 16$] #math.equation(block: true, alt: "8 open parenthesis 6 t minus 5 close parenthesis minus 35 equals −27")[$8 ( 6 t − 5 ) − 35 = −27$] #solutionbox[ #math.equation(block: true, alt: "t equals 1")[$t = 1$] ] #math.equation(block: true, alt: "3 open parenthesis 10 minus 2 x close parenthesis plus 54 equals 0")[$3 ( 10 − 2 x ) + 54 = 0$] #math.equation(block: true, alt: "−2 open parenthesis 11 minus 7 x close parenthesis plus 54 equals 4")[$−2 ( 11 − 7 x ) + 54 = 4$] #solutionbox[ #math.equation(block: true, alt: "x equals −2")[$x = −2$] ] #math.equation(block: true, alt: "the fraction 2 over 3 open parenthesis 9 c minus 3 close parenthesis equals 22")[$frac(2, 3) ( 9 c − 3 ) = 22$] #math.equation(block: true, alt: "the fraction 3 over 5 open parenthesis 10 x minus 5 close parenthesis equals 27")[$frac(3, 5) ( 10 x − 5 ) = 27$] #solutionbox[ #math.equation(block: true, alt: "x equals 5")[$x = 5$] ] #math.equation(block: true, alt: "the fraction 1 over 5 open parenthesis 15 c plus 10 close parenthesis equals c plus 7")[$frac(1, 5) ( 15 c + 10 ) = c + 7$] #math.equation(block: true, alt: "the fraction 1 over 4 open parenthesis 20 d plus 12 close parenthesis equals d plus 7")[$frac(1, 4) ( 20 d + 12 ) = d + 7$] #solutionbox[ #math.equation(block: true, alt: "d equals 1")[$d = 1$] ] #math.equation(block: true, alt: "18 minus open parenthesis 9 r plus 7 close parenthesis equals −16")[$18 − ( 9 r + 7 ) = −16$] #math.equation(block: true, alt: "15 minus open parenthesis 3 r plus 8 close parenthesis equals 28")[$15 − ( 3 r + 8 ) = 28$] #solutionbox[ #math.equation(block: true, alt: "r equals −7")[$r = −7$] ] #math.equation(block: true, alt: "5 minus open parenthesis n minus 1 close parenthesis equals 19")[$5 − ( n − 1 ) = 19$] #math.equation(block: true, alt: "−3 minus open parenthesis m minus 1 close parenthesis equals 13")[$−3 − ( m − 1 ) = 13$] #solutionbox[ #math.equation(block: true, alt: "m equals −15")[$m = −15$] ] #math.equation(block: true, alt: "11 minus 4 open parenthesis y minus 8 close parenthesis equals 43")[$11 − 4 ( y − 8 ) = 43$] #math.equation(block: true, alt: "18 minus 2 open parenthesis y minus 3 close parenthesis equals 32")[$18 − 2 ( y − 3 ) = 32$] #solutionbox[ #math.equation(block: true, alt: "y equals −4")[$y = −4$] ] #math.equation(block: true, alt: "24 minus 8 open parenthesis 3 v plus 6 close parenthesis equals 0")[$24 − 8 ( 3 v + 6 ) = 0$] #math.equation(block: true, alt: "35 minus 5 open parenthesis 2 w plus 8 close parenthesis equals −10")[$35 − 5 ( 2 w + 8 ) = −10$] #solutionbox[ #math.equation(block: true, alt: "w equals the fraction 1 over 2")[$w = frac(1, 2)$] ] #math.equation(block: true, alt: "4 open parenthesis a minus 12 close parenthesis equals 3 open parenthesis a plus 5 close parenthesis")[$4 ( a − 12 ) = 3 ( a + 5 )$] #math.equation(block: true, alt: "−2 open parenthesis a minus 6 close parenthesis equals 4 open parenthesis a minus 3 close parenthesis")[$−2 ( a − 6 ) = 4 ( a − 3 )$] #solutionbox[ #math.equation(block: true, alt: "a equals 4")[$a = 4$] ] #math.equation(block: true, alt: "2 open parenthesis 5 minus u close parenthesis equals −3 open parenthesis 2 u plus 6 close parenthesis")[$2 ( 5 − u ) = −3 ( 2 u + 6 )$] #math.equation(block: true, alt: "5 open parenthesis 8 minus r close parenthesis equals −2 open parenthesis 2 r minus 16 close parenthesis")[$5 ( 8 − r ) = −2 ( 2 r − 16 )$] #solutionbox[ #math.equation(block: true, alt: "r equals 8")[$r = 8$] ] #math.equation(block: true, alt: "3 open parenthesis 4 n minus 1 close parenthesis minus 2 equals 8 n plus 3")[$3 ( 4 n − 1 ) − 2 = 8 n + 3$] #math.equation(block: true, alt: "9 open parenthesis 2 m minus 3 close parenthesis minus 8 equals 4 m plus 7")[$9 ( 2 m − 3 ) − 8 = 4 m + 7$] #solutionbox[ #math.equation(block: true, alt: "m equals 3")[$m = 3$] ] #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")[$12 + 2 ( 5 − 3 y ) = −9 ( y − 1 ) − 2$] #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")[$−15 + 4 ( 2 − 5 y ) = −7 ( y − 4 ) + 4$] #solutionbox[ #math.equation(block: true, alt: "y equals −3")[$y = −3$] ] #math.equation(block: true, alt: "8 open parenthesis x minus 4 close parenthesis minus 7 x equals 14")[$8 ( x − 4 ) − 7 x = 14$] #math.equation(block: true, alt: "5 open parenthesis x minus 4 close parenthesis minus 4 x equals 14")[$5 ( x − 4 ) − 4 x = 14$] #solutionbox[ #math.equation(block: true, alt: "x equals 34")[$x = 34$] ] #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")[$5 + 6 ( 3 s − 5 ) = −3 + 2 ( 8 s − 1 )$] #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")[$−12 + 8 ( x − 5 ) = −4 + 3 ( 5 x − 2 )$] #solutionbox[ #math.equation(block: true, alt: "x equals −6")[$x = −6$] ] #math.equation(block: true, alt: "4 open parenthesis u minus 1 close parenthesis minus 8 equals 6 open parenthesis 3 u minus 2 close parenthesis minus 7")[$4 ( u − 1 ) − 8 = 6 ( 3 u − 2 ) − 7$] #math.equation(block: true, alt: "7 open parenthesis 2 n minus 5 close parenthesis equals 8 open parenthesis 4 n minus 1 close parenthesis minus 9")[$7 ( 2 n − 5 ) = 8 ( 4 n − 1 ) − 9$] #solutionbox[ #math.equation(block: true, alt: "n equals −1")[$n = −1$] ] #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")[$4 ( p − 4 ) − ( p + 7 ) = 5 ( p − 3 )$] #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")[$3 ( a − 2 ) − ( a + 6 ) = 4 ( a − 1 )$] #solutionbox[ #math.equation(block: true, alt: "a equals −4")[$a = −4$] ] #math.equation(block: true, alt: "− open parenthesis 9 y plus 5 close parenthesis minus open parenthesis 3 y minus 7 close parenthesis")[$"−" ( 9 y + 5 ) − ( 3 y − 7 )$] #linebreak() #math.equation(block: true, alt: "equals 16 minus open parenthesis 4 y minus 2 close parenthesis")[$= 16 − ( 4 y − 2 )$] #math.equation(block: true, alt: "− open parenthesis 7 m plus 4 close parenthesis minus open parenthesis 2 m minus 5 close parenthesis")[$"−" ( 7 m + 4 ) − ( 2 m − 5 )$] #linebreak() #math.equation(block: true, alt: "equals 14 minus open parenthesis 5 m minus 3 close parenthesis")[$= 14 − ( 5 m − 3 )$] #solutionbox[ #math.equation(block: true, alt: "m equals −4")[$m = −4$] ] #math.equation(block: true, alt: "4 [ 5 minus 8 open parenthesis 4 c minus 3 close parenthesis ]")[$4 [ 5 − 8 ( 4 c − 3 ) ]$] #linebreak() #math.equation(block: true, alt: "equals 12 open parenthesis 1 minus 13 c close parenthesis minus 8")[$= 12 ( 1 − 13 c ) − 8$] #math.equation(block: true, alt: "5 [ 9 minus 2 open parenthesis 6 d minus 1 close parenthesis ]")[$5 [ 9 − 2 ( 6 d − 1 ) ]$] #linebreak() #math.equation(block: true, alt: "equals 11 open parenthesis 4 minus 10 d close parenthesis minus 139")[$= 11 ( 4 − 10 d ) − 139$] #solutionbox[ #math.equation(block: true, alt: "d equals −3")[$d = −3$] ] #math.equation(block: true, alt: "3 [ −9 plus 8 open parenthesis 4 h minus 3 close parenthesis ]")[$3 [ −9 + 8 ( 4 h − 3 ) ]$] #linebreak() #math.equation(block: true, alt: "equals 2 open parenthesis 5 minus 12 h close parenthesis minus 19")[$= 2 ( 5 − 12 h ) − 19$] #math.equation(block: true, alt: "3 [ −14 plus 2 open parenthesis 15 k minus 6 close parenthesis ]")[$3 [ −14 + 2 ( 15 k − 6 ) ]$] #linebreak() #math.equation(block: true, alt: "equals 8 open parenthesis 3 minus 5 k close parenthesis minus 24")[$= 8 ( 3 − 5 k ) − 24$] #solutionbox[ #math.equation(block: true, alt: "k equals the fraction 3 over 5")[$k = frac(3, 5)$] ] #math.equation(block: true, alt: "5 [ 2 open parenthesis m plus 4 close parenthesis plus 8 open parenthesis m minus 7 close parenthesis ]")[$5 [ 2 ( m + 4 ) + 8 ( m − 7 ) ]$] #linebreak() #math.equation(block: true, alt: "equals 2 [ 3 open parenthesis 5 plus m close parenthesis minus open parenthesis 21 minus 3 m close parenthesis ]")[$= 2 [ 3 ( 5 + m ) − ( 21 − 3 m ) ]$] #math.equation(block: true, alt: "10 [ 5 open parenthesis n plus 1 close parenthesis plus 4 open parenthesis n minus 1 close parenthesis ]")[$10 [ 5 ( n + 1 ) + 4 ( n − 1 ) ]$] #linebreak() #math.equation(block: true, alt: "equals 11 [ 7 open parenthesis 5 plus n close parenthesis minus open parenthesis 25 minus 3 n close parenthesis ]")[$= 11 [ 7 ( 5 + n ) − ( 25 − 3 n ) ]$] #solutionbox[ #math.equation(block: true, alt: "n equals −5")[$n = −5$] ] #math.equation(block: true, alt: "5 open parenthesis 1.2 u minus 4.8 close parenthesis equals −12")[$5 ( 1.2 u − 4.8 ) = −12$] #math.equation(block: true, alt: "4 open parenthesis 2.5 v minus 0.6 close parenthesis equals 7.6")[$4 ( 2.5 v − 0.6 ) = 7.6$] #solutionbox[ #math.equation(block: true, alt: "v equals 1")[$v = 1$] ] #math.equation(block: true, alt: "0.25 open parenthesis q minus 6 close parenthesis equals 0.1 open parenthesis q plus 18 close parenthesis")[$0.25 ( q − 6 ) = 0.1 ( q + 18 )$] #math.equation(block: true, alt: "0.2 open parenthesis p minus 6 close parenthesis equals 0.4 open parenthesis p plus 14 close parenthesis")[$0.2 ( p − 6 ) = 0.4 ( p + 14 )$] #solutionbox[ #math.equation(block: true, alt: "p equals −34")[$p = −34$] ] #math.equation(block: true, alt: "0.2 open parenthesis 30 n plus 50 close parenthesis equals 28")[$0.2 ( 30 n + 50 ) = 28$] #math.equation(block: true, alt: "0.5 open parenthesis 16 m plus 34 close parenthesis equals −15")[$0.5 ( 16 m + 34 ) = −15$] #solutionbox[ #math.equation(block: true, alt: "m equals −4")[$m = −4$] ] #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")[$23 z + 19 = 3 ( 5 z − 9 ) + 8 z + 46$] #math.equation(block: true, alt: "15 y plus 32 equals 2 open parenthesis 10 y minus 7 close parenthesis minus 5 y plus 46")[$15 y + 32 = 2 ( 10 y − 7 ) − 5 y + 46$] #solutionbox[ identity; all real numbers ] #math.equation(block: true, alt: "5 open parenthesis b minus 9 close parenthesis plus 4 open parenthesis 3 b plus 9 close parenthesis equals 6 open parenthesis 4 b minus 5 close parenthesis minus 7 b plus 21")[$5 ( b − 9 ) + 4 ( 3 b + 9 ) = 6 ( 4 b − 5 ) − 7 b + 21$] #math.equation(block: true, alt: "9 open parenthesis a minus 4 close parenthesis plus 3 open parenthesis 2 a plus 5 close parenthesis equals 7 open parenthesis 3 a minus 4 close parenthesis minus 6 a plus 7")[$9 ( a − 4 ) + 3 ( 2 a + 5 ) = 7 ( 3 a − 4 ) − 6 a + 7$] #solutionbox[ identity; all real numbers ] #math.equation(block: true, alt: "18 open parenthesis 5 j minus 1 close parenthesis plus 29 equals 47")[$18 ( 5 j − 1 ) + 29 = 47$] #math.equation(block: true, alt: "24 open parenthesis 3 d minus 4 close parenthesis plus 100 equals 52")[$24 ( 3 d − 4 ) + 100 = 52$] #solutionbox[ conditional equation; #math.equation(block: false, alt: "d equals the fraction 2 over 3")[$d = frac(2, 3)$] ] #math.equation(block: true, alt: "22 open parenthesis 3 m minus 4 close parenthesis equals 8 open parenthesis 2 m plus 9 close parenthesis")[$22 ( 3 m − 4 ) = 8 ( 2 m + 9 )$] #math.equation(block: true, alt: "30 open parenthesis 2 n minus 1 close parenthesis equals 5 open parenthesis 10 n plus 8 close parenthesis")[$30 ( 2 n − 1 ) = 5 ( 10 n + 8 )$] #solutionbox[ conditional equation; #math.equation(block: false, alt: "n equals 7")[$n = 7$] ] #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")[$7 v + 42 = 11 ( 3 v + 8 ) − 2 ( 13 v − 1 )$] #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")[$18 u − 51 = 9 ( 4 u + 5 ) − 6 ( 3 u − 10 )$] #solutionbox[ contradiction; no solution ] #math.equation(block: true, alt: "3 open parenthesis 6 q minus 9 close parenthesis plus 7 open parenthesis q plus 4 close parenthesis equals 5 open parenthesis 6 q plus 8 close parenthesis minus 5 open parenthesis q plus 1 close parenthesis")[$3 ( 6 q − 9 ) + 7 ( q + 4 ) = 5 ( 6 q + 8 ) − 5 ( q + 1 )$] #math.equation(block: true, alt: "5 open parenthesis p plus 4 close parenthesis plus 8 open parenthesis 2 p minus 1 close parenthesis equals 9 open parenthesis 3 p minus 5 close parenthesis minus 6 open parenthesis p minus 2 close parenthesis")[$5 ( p + 4 ) + 8 ( 2 p − 1 ) = 9 ( 3 p − 5 ) − 6 ( p − 2 )$] #solutionbox[ contradiction; no solution ] #math.equation(block: true, alt: "12 open parenthesis 6 h minus 1 close parenthesis equals 8 open parenthesis 8 h plus 5 close parenthesis minus 4")[$12 ( 6 h − 1 ) = 8 ( 8 h + 5 ) − 4$] #math.equation(block: true, alt: "9 open parenthesis 4 k minus 7 close parenthesis equals 11 open parenthesis 3 k plus 1 close parenthesis plus 4")[$9 ( 4 k − 7 ) = 11 ( 3 k + 1 ) + 4$] #solutionbox[ conditional equation; #math.equation(block: false, alt: "k equals 26")[$k = 26$] ] #math.equation(block: true, alt: "45 open parenthesis 3 y minus 2 close parenthesis equals 9 open parenthesis 15 y minus 6 close parenthesis")[$45 ( 3 y − 2 ) = 9 ( 15 y − 6 )$] #math.equation(block: true, alt: "60 open parenthesis 2 x minus 1 close parenthesis equals 15 open parenthesis 8 x plus 5 close parenthesis")[$60 ( 2 x − 1 ) = 15 ( 8 x + 5 )$] #solutionbox[ contradiction; no solution ] #math.equation(block: true, alt: "16 open parenthesis 6 n plus 15 close parenthesis equals 48 open parenthesis 2 n plus 5 close parenthesis")[$16 ( 6 n + 15 ) = 48 ( 2 n + 5 )$] #math.equation(block: true, alt: "36 open parenthesis 4 m plus 5 close parenthesis equals 12 open parenthesis 12 m plus 15 close parenthesis")[$36 ( 4 m + 5 ) = 12 ( 12 m + 15 )$] #solutionbox[ identity; all real numbers ] #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")[$9 ( 14 d + 9 ) + 4 d = 13 ( 10 d + 6 ) + 3$] #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")[$11 ( 8 c + 5 ) − 8 c = 2 ( 40 c + 25 ) + 5$] #solutionbox[ identity; all real numbers ] ==== Everyday Math #strong[Fencing] Micah has 44 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 44")[$2 L + 2 ( L − 2.5 ) = 44$]. #strong[Coins] Rhonda has \$1.90 in nickels and dimes. The number of dimes is one less than twice the number of nickels. Find the number of nickels, #emph[n], by solving the equation #math.equation(block: false, alt: "0.05 n plus 0.10 open parenthesis 2 n minus 1 close parenthesis equals 1.90")[$0.05 n + 0.10 ( 2 n − 1 ) = 1.90$]. #solutionbox[ 8 nickels ] ==== Writing Exercises Using your own words, list the steps in the general strategy for solving linear equations. 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. #solutionbox[ Answers will vary. ] 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? Solve the equation #math.equation(block: false, alt: "the fraction 1 over 4 open parenthesis 8 x plus 20 close parenthesis equals 3 x minus 4")[$frac(1, 4) ( 8 x + 20 ) = 3 x − 4$] explaining all the steps of your solution as in the examples in this section. #solutionbox[ Answers will vary. ] ==== Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objective of this section. #figure(figph[This is a table that has three rows and four columns. In the first row, which is a header row, the cells read from left to right: “I can…,” “confidently,” “with some help,” and “no-I don’t get it!” The first column below “I can…” reads: “solve equations using the general strategy for solving linear equations,” and “classify equations.” The rest of the cells are blank.], alt: "This is a table that has three rows and four columns. In the first row, which is a header row, the cells read from left to right: “I can…,” “confidently,” “with some help,” and “no-I don’t get it!” The first column below “I can…” reads: “solve equations using the general strategy for solving linear equations,” and “classify equations.” The rest of the cells are blank.", caption: none) ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?