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== 2.5#h(0.6em)Solve Equations with Fractions or Decimals

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Before you get started, take this readiness quiz.

Multiply: #math.equation(block: false, alt: "8 times the fraction 3 over 8 .")[$8 · frac(3, 8) .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "3")[$3$]
]
]

#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")[$12$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply 4.78 by 100. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "478")[$478$]
]
]


=== Solve Equations with Fraction Coefficients

Let’s use the general strategy for solving linear equations introduced earlier to solve the equation, #math.equation(block: false, alt: "the fraction 1 over 8 x plus the fraction 1 over 2 equals the fraction 1 over 4")[$frac(1, 8) x + frac(1, 2) = frac(1, 4)$].

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed: 1/8x + 1/2 = 1/4. The equation involves fractions, a variable 'x', addition, and an equality sign, set against a plain white background.], alt: "A mathematical equation is displayed: 1/8x + 1/2 = 1/4. The equation involves fractions, a variable 'x', addition, and an equality sign, set against a plain white background.", caption: none)]),
  [To isolate the #math.equation(block: false, alt: "x")[$x$] term, subtract #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$] from both sides.], [#figure(figph[A mathematical equation displays '1/8x + 1/2 - 1/2 = 1/4 - 1/2'. The second '1/2' on the left side of the equation is shown in red and has a dashed line through it, as does the '1/2' on the right side.], alt: "A mathematical equation displays '1/8x + 1/2 - 1/2 = 1/4 - 1/2'. The second '1/2' on the left side of the equation is shown in red and has a dashed line through it, as does the '1/2' on the right side.", caption: none)],
  [Simplify the left side.], [#figure(figph[A mathematical equation is displayed, showing one eighth times x equals one fourth minus one half: (1/8)x = (1/4) - (1/2).], alt: "A mathematical equation is displayed, showing one eighth times x equals one fourth minus one half: (1/8)x = (1/4) - (1/2).", caption: none)],
  [Change the constants to equivalent fractions with the LCD.], [#figure(figph[A mathematical equation is displayed, showing '1/8 x = 1/4 - 2/4' on a white background.], alt: "A mathematical equation is displayed, showing '1/8 x = 1/4 - 2/4' on a white background.", caption: none)],
  [Subtract.], [#figure(figph[A mathematical equation shows one-eighth multiplied by x, set equal to negative one-fourth: (1/8)x = -1/4.], alt: "A mathematical equation shows one-eighth multiplied by x, set equal to negative one-fourth: (1/8)x = -1/4.", caption: none)],
  [Multiply both sides by the reciprocal of #math.equation(block: false, alt: "the fraction 1 over 8")[$frac(1, 8)$].], [#figure(figph[A mathematical equation is displayed, showing 8/1 multiplied by 1/8x, which equals 8/1 multiplied by -1/4.], alt: "A mathematical equation is displayed, showing 8/1 multiplied by 1/8x, which equals 8/1 multiplied by -1/4.", caption: none)],
  [Simplify.], [#figure(figph[The image shows the equation 'x = -2' written in a simple, clear font against a white background.], alt: "The image shows the equation 'x = -2' written in a simple, clear font against a white background.", caption: none)],
))

This method worked 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 least common denominator 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 “clearing” the equation of fractions.

Let’s solve a similar equation, but this time use the method that eliminates the fractions.

#examplebox("Example 1")[How to Solve Equations with Fraction Coefficients][
Solve: #math.equation(block: false, alt: "the fraction 1 over 6 y minus the fraction 1 over 3 equals the fraction 5 over 6")[$frac(1, 6) y − frac(1, 3) = frac(5, 6)$].

#solutionbox[
#figure(figph[This figure is a table that has three columns and three 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. Find the least common denominator of all the fractions in the equation.” The text in the second cell reads: “What is the LCD of 1/6, 1/3, and 5/6?” The third cell contains the equation one-sixth y minus 1/3 equals 5/6, with LCD equals 6 written next to it.], alt: "This figure is a table that has three columns and three 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. Find the least common denominator of all the fractions in the equation.” The text in the second cell reads: “What is the LCD of 1/6, 1/3, and 5/6?” The third cell contains the equation one-sixth y minus 1/3 equals 5/6, with LCD equals 6 written next to it.", caption: none)

#figure(figph[In the second row of the table, the first cell says: “Step 2. Multiply both sides of the equation by that LCD. This clears the fractions.” In the second cell, the instructions say: “Multiply both sides of the equation by the LCD 6. Use the Distributive Property. Simplify—and notice, no more fractions!” The third cell contains the equation 6 times one-sixth y minus 1/3, with one-sixth y minus 1/3 in brackets, equals 6 times 5/6, with “6 times” written in red on both sides. Below this is the same equation with the 6 distributed on both sides: 6 times one-sixth y minus 6 times 1/3 equals 6 times 5/6. Below this is the equation y minus 2 equals 5.], alt: "In the second row of the table, the first cell says: “Step 2. Multiply both sides of the equation by that LCD. This clears the fractions.” In the second cell, the instructions say: “Multiply both sides of the equation by the LCD 6. Use the Distributive Property. Simplify—and notice, no more fractions!” The third cell contains the equation 6 times one-sixth y minus 1/3, with one-sixth y minus 1/3 in brackets, equals 6 times 5/6, with “6 times” written in red on both sides. Below this is the same equation with the 6 distributed on both sides: 6 times one-sixth y minus 6 times 1/3 equals 6 times 5/6. Below this is the equation y minus 2 equals 5.", caption: none)

#figure(figph[In the third row of the table, the first cell says: “Step 3. Solve using the General Strategy for Solving Linear Equations.” In the second cell, the instructions say: “Isolate the x term, add 2. Simplify.” The third cell contains the equation with 2 added to both sides: y minus 2 plus 2 equals 5 plus 2, with “plus 2” written in red on both sides. Below this is the equation y equals 7.], alt: "In the third row of the table, the first cell says: “Step 3. Solve using the General Strategy for Solving Linear Equations.” In the second cell, the instructions say: “Isolate the x term, add 2. Simplify.” The third cell contains the equation with 2 added to both sides: y minus 2 plus 2 equals 5 plus 2, with “plus 2” written in red on both sides. Below this is the equation y equals 7.", 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 , 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 General Strategy for Solving Linear Equations.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Strategy to solve equations with fraction coefficients.]

+ Find the least common denominator of #emph[all] the fractions in the equation.
+ Multiply both sides of the equation by that LCD. This clears the fractions.
+ Solve using the General Strategy for Solving Linear Equations.
]

#examplebox("Example 2")[][
Solve: #math.equation(block: false, alt: "6 equals the fraction 1 over 2 v plus the fraction 2 over 5 v minus the fraction 3 over 4 v")[$6 = frac(1, 2) v + frac(2, 5) v − frac(3, 4) v$].

#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 shown is 6 = 1/2v + 2/5v - 3/4v, displaying a linear equation with fractions where the constant 6 is set equal to a sum and difference of terms involving the variable 'v' with fractional coefficients.], alt: "A mathematical equation shown is 6 = 1/2v + 2/5v - 3/4v, displaying a linear equation with fractions where the constant 6 is set equal to a sum and difference of terms involving the variable 'v' with fractional coefficients.", caption: none)], [],
  [The LCD is 20.], [], [],
  [Multiply both sides of the equation by 20.], [#figure(figph[A mathematical equation shows '20(6) = 20 \* (1/2 v + 2/5 v - 3/4 v)' with the number 20 highlighted in red on both sides of the equals sign.], alt: "A mathematical equation shows '20(6) = 20 * (1/2 v + 2/5 v - 3/4 v)' with the number 20 highlighted in red on both sides of the equals sign.", caption: none)], [],
  [Distribute.], [#figure(figph[A mathematical equation shows 20 multiplied by 6 on the left side, equaling the sum of three terms on the right: 20 times 1/2v, plus 20 times 2/5v, minus 20 times 3/4v.], alt: "A mathematical equation shows 20 multiplied by 6 on the left side, equaling the sum of three terms on the right: 20 times 1/2v, plus 20 times 2/5v, minus 20 times 3/4v.", caption: none)], [],
  [Simplify—notice, no more fractions!], [#figure(figph[An image displays the algebraic equation 120 = 10v + 8v - 15v, where a numerical value is equated to a sum and difference of terms involving the variable 'v'.], alt: "An image displays the algebraic equation 120 = 10v + 8v - 15v, where a numerical value is equated to a sum and difference of terms involving the variable 'v'.", caption: none)], [],
  [Combine like terms.], [#figure(figph[A mathematical equation is displayed on a white background, showing '120 = 3y' in a simple, gray font. This is a linear equation with one variable.], alt: "A mathematical equation is displayed on a white background, showing '120 = 3y' in a simple, gray font. This is a linear equation with one variable.", caption: none)], [],
  [Divide by 3.], [#figure(figph[A mathematical equation shown as 120 divided by 3 equals 3v divided by 3, with the number 3 on the denominators highlighted in red, indicating a division step.], alt: "A mathematical equation shown as 120 divided by 3 equals 3v divided by 3, with the number 3 on the denominators highlighted in red, indicating a division step.", caption: none)], [],
  [Simplify.], [#figure(figph[The equation 40 = V is displayed on a white background.], alt: "The equation 40 = V is displayed on a white background.", caption: none)], [],
  [Check:], [#figure(figph[A mathematical equation is displayed: 6 = (1/2)v + (2/5)v - (3/4)v.], alt: "A mathematical equation is displayed: 6 = (1/2)v + (2/5)v - (3/4)v.", caption: none)], [],
  [Let #math.equation(block: false, alt: "v equals 40")[$v = 40$].], [#figure(figph[A mathematical equation where 6 is questioned to be equal to one-half of 40 plus two-fifths of 40 minus three-fourths of 40.], alt: "A mathematical equation where 6 is questioned to be equal to one-half of 40 plus two-fifths of 40 minus three-fourths of 40.", caption: none)], [],
  [], [#figure(figph[A mathematical equation checks if 6 is equal to 20 + 16 - 30, which simplifies to 6. The equation is presented as '6 ?= 20 + 16 - 30' against a white background.], alt: "A mathematical equation checks if 6 is equal to 20 + 16 - 30, which simplifies to 6. The equation is presented as '6 ?= 20 + 16 - 30' against a white background.", caption: none)], [],
  [], [#figure(figph[The mathematical expression '6 = 6√' is shown on a white background.], alt: "The mathematical expression '6 = 6√' is shown 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 again have variables on both sides of the equation.

#examplebox("Example 3")[][
Solve: #math.equation(block: false, alt: "a plus the fraction 3 over 4 equals the fraction 3 over 8 a minus the fraction 1 over 2")[$a + frac(3, 4) = frac(3, 8) a − frac(1, 2)$].

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[A mathematical equation is displayed: a + 3/4 = 3/8 a - 1/2.], alt: "A mathematical equation is displayed: a + 3/4 = 3/8 a - 1/2.", caption: none)], [],
  [Find the LCD of all fractions in the equation. #linebreak() The LCD is 8.], [], [],
  [Multiply both sides by the LCD.], [#figure(figph[A mathematical equation is displayed: 8(a + 3/4) = 8(3/8a - 1/2). The equation involves a variable 'a', integers, and fractions, with both sides multiplied by the integer 8.], alt: "A mathematical equation is displayed: 8(a + 3/4) = 8(3/8a - 1/2). The equation involves a variable 'a', integers, and fractions, with both sides multiplied by the integer 8.", caption: none)], [],
  [Distribute.], [#figure(figph[A mathematical equation displays 8 multiplied by 'a' plus 8 multiplied by three-fourths equals 8 multiplied by three-eighths multiplied by 'a' minus 8 multiplied by one-half.], alt: "A mathematical equation displays 8 multiplied by 'a' plus 8 multiplied by three-fourths equals 8 multiplied by three-eighths multiplied by 'a' minus 8 multiplied by one-half.", caption: none)], [],
  [Simplify—no more fractions.], [#figure(figph[An algebraic equation is shown on a white background, reading '8a + 6 = 3a - 4'.], alt: "An algebraic equation is shown on a white background, reading '8a + 6 = 3a - 4'.", caption: none)], [],
  [Subtract #math.equation(block: false, alt: "3 a")[$3 a$] from both sides.], [#figure(figph[An algebraic equation displayed as 8a - 3a + 6 = 3a - 3a - 4, where the terms '-3a' and '3a' appear in red.], alt: "An algebraic equation displayed as 8a - 3a + 6 = 3a - 3a - 4, where the terms '-3a' and '3a' appear in red.", caption: none)], [],
  [Simplify.], [#figure(figph[A mathematical equation is displayed, reading '5a + 6 = -4' in a grayscale text against a white background.], alt: "A mathematical equation is displayed, reading '5a + 6 = -4' in a grayscale text against a white background.", caption: none)], [],
  [Subtract 6 from both sides.], [#figure(figph[A mathematical equation is displayed on a white background: 5a + 6 - 6 = -4 - 6. The subtracted 6s on both sides of the equation are highlighted in red.], alt: "A mathematical equation is displayed on a white background: 5a + 6 - 6 = -4 - 6. The subtracted 6s on both sides of the equation are highlighted in red.", caption: none)], [],
  [Simplify.], [#figure(figph[A mathematical equation is displayed on a white background, reading '5q = -10'. The numbers and variable are in a dark gray font.], alt: "A mathematical equation is displayed on a white background, reading '5q = -10'. The numbers and variable are in a dark gray font.", caption: none)], [],
  [Divide by 5.], [#figure(figph[A mathematical equation shows '5a divided by 5 equals -10 divided by 5'. The number 5 in the denominator of both fractions is highlighted in red, indicating division by 5 on both sides of the equation.], alt: "A mathematical equation shows '5a divided by 5 equals -10 divided by 5'. The number 5 in the denominator of both fractions is highlighted in red, indicating division by 5 on both sides of the equation.", caption: none)], [],
  [Simplify.], [#figure(figph[The image displays the equation 'q = -2' in black text against a plain white background.], alt: "The image displays the equation 'q = -2' in black text against a plain white background.", caption: none)], [],
  [Check:], [#figure(figph[An algebraic equation showing 'a plus three-fourths equals three-eighths 'a' minus one-half'.], alt: "An algebraic equation showing 'a plus three-fourths equals three-eighths 'a' minus one-half'.", caption: none)], [],
  [Let #math.equation(block: false, alt: "a equals −2")[$a = −2$].], [#figure(figph[A mathematical problem presented as an equation to be verified: -2 + 3/4 =? (3/8)(-2) - 1/2, involving integers and fractions.], alt: "A mathematical problem presented as an equation to be verified: -2 + 3/4 =? (3/8)(-2) - 1/2, involving integers and fractions.", caption: none)], [],
  [], [#figure(figph[A mathematical equation featuring fractions: -8/4 + 3/4 = ? 16/8 - 4/8. The question mark indicates an unknown value or a query regarding the equality of the expressions.], alt: "A mathematical equation featuring fractions: -8/4 + 3/4 = ? 16/8 - 4/8. The question mark indicates an unknown value or a query regarding the equality of the expressions.", caption: none)], [],
  [], [#figure(figph[A mathematical equation shows that the negative fraction -5/4 is equal to the negative fraction -10/8, demonstrating the concept of equivalent fractions with negative values.], alt: "A mathematical equation shows that the negative fraction -5/4 is equal to the negative fraction -10/8, demonstrating the concept of equivalent fractions with negative values.", caption: none)], [],
  [], [#figure(figph[A mathematical equation shows '-5/4 = -5/4' with a checkmark, indicating the equality is correct. The expression represents a negative fraction equal to itself, often used in math education.], alt: "A mathematical equation shows '-5/4 = -5/4' with a checkmark, indicating the equality is correct. The expression represents a negative fraction equal to itself, often used in math education.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "x plus the fraction 1 over 3 equals the fraction 1 over 6 x minus the fraction 1 over 2")[$x + frac(1, 3) = frac(1, 6) x − frac(1, 2)$].

#solutionbox[
#math.equation(block: true, alt: "x equals −1")[$x = −1$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "c plus the fraction 3 over 4 equals the fraction 1 over 2 c minus the fraction 1 over 4")[$c + frac(3, 4) = frac(1, 2) c − frac(1, 4)$].

#solutionbox[
#math.equation(block: true, alt: "c equals −2")[$c = −2$]
]
]

In the next example, we start by using the Distributive Property. This step clears the fractions right away.

#examplebox("Example 4")[][
Solve: #math.equation(block: false, alt: "−5 equals the fraction 1 over 4 open parenthesis 8 x plus 4 close parenthesis")[$−5 = frac(1, 4) ( 8 x + 4 )$].

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[A mathematical equation is displayed against a white background: -5 = 1/4 (8x + 4).], alt: "A mathematical equation is displayed against a white background: -5 = 1/4 (8x + 4).", caption: none)], [],
  [Distribute.], [#figure(figph[An algebraic equation is shown, displaying -5 = 1/4 \* 8x + 1/4 \* 4.], alt: "An algebraic equation is shown, displaying -5 = 1/4 * 8x + 1/4 * 4.", caption: none)], [],
  [Simplify. #linebreak() Now there are no fractions.], [#figure(figph[A clear image displaying the mathematical equation '-5 = 2x + 1' centered against a white background.], alt: "A clear image displaying the mathematical equation '-5 = 2x + 1' centered against a white background.", caption: none)], [],
  [Subtract 1 from both sides.], [#figure(figph[A mathematical equation is displayed on a white background: -5 - 1 = 2x + 1 - 1. The '-1' terms are highlighted in red on both sides, indicating a step in solving the equation.], alt: "A mathematical equation is displayed on a white background: -5 - 1 = 2x + 1 - 1. The '-1' terms are highlighted in red on both sides, indicating a step in solving the equation.", caption: none)], [],
  [Simplify.], [#figure(figph[A basic algebra problem showing the equation -6 = 2x is presented on a white background, demonstrating a straightforward linear equation to solve for x.], alt: "A basic algebra problem showing the equation -6 = 2x is presented on a white background, demonstrating a straightforward linear equation to solve for x.", caption: none)], [],
  [Divide by 2.], [#figure(figph[A mathematical equation shows '-6 divided by 2 equals 2x divided by 2.' The denominator '2' on both sides of the equation is highlighted in red.], alt: "A mathematical equation shows '-6 divided by 2 equals 2x divided by 2.' The denominator '2' on both sides of the equation is highlighted in red.", caption: none)], [],
  [Simplify.], [#figure(figph[The mathematical equation -3 = X is displayed against a plain white background, showing the solution for X.], alt: "The mathematical equation -3 = X is displayed against a plain white background, showing the solution for X.", caption: none)], [],
  [Check:], [#figure(figph[A mathematical equation is displayed on a white background: -5 = 1/4(8x + 4). The equation shows -5 on the left side of the equals sign. On the right, the fraction 1/4 is multiplied by the quantity (8x + 4).], alt: "A mathematical equation is displayed on a white background: -5 = 1/4(8x + 4). The equation shows -5 on the left side of the equals sign. On the right, the fraction 1/4 is multiplied by the quantity (8x + 4).", caption: none)], [],
  [Let #math.equation(block: false, alt: "x equals −3")[$x = −3$].], [#figure(figph[A mathematical equation reads '-5 =? (1/2) (4(-3) + 2)' on a white background. The number -3 is highlighted in red, indicating a point of interest or a specific value being evaluated.], alt: "A mathematical equation reads '-5 =? (1/2) (4(-3) + 2)' on a white background. The number -3 is highlighted in red, indicating a point of interest or a specific value being evaluated.", caption: none)], [],
  [], [#figure(figph[A mathematical equation is displayed on a white background: -5 =? 1/2(-12 + 2). The equation asks if -5 is equal to half of the sum of -12 and 2.], alt: "A mathematical equation is displayed on a white background: -5 =? 1/2(-12 + 2). The equation asks if -5 is equal to half of the sum of -12 and 2.", caption: none)], [],
  [], [#figure(figph[A math problem asks whether -5 is equal to (1/2) \* (-10). The equation is true as (1/2) \* (-10) simplifies to -5.], alt: "A math problem asks whether -5 is equal to (1/2) * (-10). The equation is true as (1/2) * (-10) simplifies to -5.", caption: none)], [],
  [], [#figure(figph[A mathematical equation shows '-5 = -5' followed by a checkmark, indicating that the statement is correct.], alt: "A mathematical equation shows '-5 = -5' followed by a checkmark, indicating that the statement is correct.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "−11 equals the fraction 1 over 2 open parenthesis 6 p plus 2 close parenthesis")[$−11 = frac(1, 2) ( 6 p + 2 )$].

#solutionbox[
#math.equation(block: true, alt: "p equals −4")[$p = −4$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "8 equals the fraction 1 over 3 open parenthesis 9 q plus 6 close parenthesis")[$8 = frac(1, 3) ( 9 q + 6 )$].

#solutionbox[
#math.equation(block: true, alt: "q equals 2")[$q = 2$]
]
]

In the next example, even after distributing, we still have fractions to clear.

#examplebox("Example 5")[][
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: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed on a white background: (1/2)(y-5) = (1/4)(y-1).], alt: "A mathematical equation is displayed on a white background: (1/2)(y-5) = (1/4)(y-1).", caption: none)]),
  [Distribute.], [#figure(figph[The image shows a linear equation involving fractions and the variable 'y': (1/2) \* y - (1/2) \* 5 = (1/4) \* y - (1/4) \* 1.], alt: "The image shows a linear equation involving fractions and the variable 'y': (1/2) * y - (1/2) * 5 = (1/4) * y - (1/4) * 1.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation is displayed on a white background: (1/2)y - (5/2) = (1/4)y - (1/4).], alt: "A mathematical equation is displayed on a white background: (1/2)y - (5/2) = (1/4)y - (1/4).", caption: none)],
  [Multiply by the LCD, 4.], [#figure(figph[A mathematical equation displays 4 multiplied by the quantity of one-half y minus five-halves, equaling 4 multiplied by the quantity of one-fourth y minus one-fourth.], alt: "A mathematical equation displays 4 multiplied by the quantity of one-half y minus five-halves, equaling 4 multiplied by the quantity of one-fourth y minus one-fourth.", caption: none)],
  [Distribute.], [#figure(figph[A linear equation involving fractions and the variable 'y' is presented as 4(1/2)y - 4(5/2) = 4(1/4)y - 4(1/4).], alt: "A linear equation involving fractions and the variable 'y' is presented as 4(1/2)y - 4(5/2) = 4(1/4)y - 4(1/4).", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation is displayed, showing 2y - 10 = y - 1, which can be solved for the variable 'y'.], alt: "A mathematical equation is displayed, showing 2y - 10 = y - 1, which can be solved for the variable 'y'.", caption: none)],
  [Collect the variables to the left.], [#figure(figph[A mathematical equation is displayed with some terms in red: '2y - y - 10 = y - y - 1'.], alt: "A mathematical equation is displayed with some terms in red: '2y - y - 10 = y - y - 1'.", caption: none)],
  [Simplify.], [#figure(figph[The equation y - 10 = -1 is displayed, representing a simple algebraic problem to solve for the variable y.], alt: "The equation y - 10 = -1 is displayed, representing a simple algebraic problem to solve for the variable y.", caption: none)],
  [Collect the constants to the right.], [#figure(figph[A step in solving an algebraic equation, demonstrating how 10 is added to both sides of y - 10 = -1 to isolate the variable y, resulting in y = 9.], alt: "A step in solving an algebraic equation, demonstrating how 10 is added to both sides of y - 10 = -1 to isolate the variable y, resulting in y = 9.", caption: none)],
  [Simplify.], [#figure(figph[The image displays the equation 'y = 9' in black text, centrally positioned on a plain white background.], alt: "The image displays the equation 'y = 9' in black text, centrally positioned on a plain white background.", caption: none)],
  [Check:], [#figure(figph[A mathematical equation is displayed on a white background. It shows '1/2(y - 5) = 1/4(y - 1)', representing a linear equation to be solved for the variable 'y'.], alt: "A mathematical equation is displayed on a white background. It shows '1/2(y - 5) = 1/4(y - 1)', representing a linear equation to be solved for the variable 'y'.", caption: none)],
  [Let #math.equation(block: false, alt: "y equals 9")[$y = 9$].], [#figure(figph[A mathematical equation asks whether (1/2)(9-5) is equal to (1/4)(9-1).], alt: "A mathematical equation asks whether (1/2)(9-5) is equal to (1/4)(9-1).", 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")[$m = −1$]
]
]

#examplebox("Example 6")[][
Solve: #math.equation(block: false, alt: "the fraction 5 x minus 3 over 4 equals the fraction x over 2")[$frac(5 x − 3, 4) = frac(x, 2)$].

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[An algebraic equation is shown, with a fraction on each side of the equals sign. On the left, the numerator is 5x - 3 and the denominator is 4. On the right, the numerator is x and the denominator is 2.], alt: "An algebraic equation is shown, with a fraction on each side of the equals sign. On the left, the numerator is 5x - 3 and the denominator is 4. On the right, the numerator is x and the denominator is 2.", caption: none)], [],
  [Multiply by the LCD, 4.], [#figure(figph[A mathematical equation shows '4 multiplied by the fraction (5x minus 3) over 4' equals '4 multiplied by the fraction x over 2.' The number 4 is highlighted in red on both sides of the equation.], alt: "A mathematical equation shows '4 multiplied by the fraction (5x minus 3) over 4' equals '4 multiplied by the fraction x over 2.' The number 4 is highlighted in red on both sides of the equation.", caption: none)], [],
  [Simplify.], [#figure(figph[A mathematical equation on a white background, '5x - 3 = 2x'. It's a linear equation in algebra, where the goal is to solve for the variable x.], alt: "A mathematical equation on a white background, '5x - 3 = 2x'. It's a linear equation in algebra, where the goal is to solve for the variable x.", caption: none)], [],
  [Collect the variables to the right.], [#figure(figph[A mathematical equation displayed as 5x - 5x - 3 = 2x - 5x, featuring variables and numbers with certain terms highlighted in red.], alt: "A mathematical equation displayed as 5x - 5x - 3 = 2x - 5x, featuring variables and numbers with certain terms highlighted in red.", caption: none)], [],
  [Simplify.], [#figure(figph[A mathematical equation showing -3 = -3x, where the solution for x is 1.], alt: "A mathematical equation showing -3 = -3x, where the solution for x is 1.", caption: none)], [],
  [Divide.], [#figure(figph[A mathematical equation showing -3 divided by -3 equals -3x divided by -3, with the denominator -3 highlighted in red.], alt: "A mathematical equation showing -3 divided by -3 equals -3x divided by -3, with the denominator -3 highlighted in red.", caption: none)], [],
  [Simplify.], [#figure(figph[The image shows a mathematical equation '1 = x' in a simple, clear font on a white background.], alt: "The image shows a mathematical equation '1 = x' in a simple, clear font on a white background.", caption: none)], [],
  [Check:], [#figure(figph[A mathematical equation is shown, with (5x - 3) divided by 4 on the left side of the equals sign, and x divided by 2 on the right side.], alt: "A mathematical equation is shown, with (5x - 3) divided by 4 on the left side of the equals sign, and x divided by 2 on the right side.", caption: none)], [],
  [Let #math.equation(block: false, alt: "x equals 1")[$x = 1$].], [#figure(figph[A mathematical equation is presented, questioning if (5(1) - 3) / 4 equals 1/2, with the '1' highlighted in red within the parentheses.], alt: "A mathematical equation is presented, questioning if (5(1) - 3) / 4 equals 1/2, with the '1' highlighted in red within the parentheses.", caption: none)], [],
  [], [#figure(figph[A mathematical equation questions whether the fraction 2/4 is equivalent to 1/2, illustrating a common concept in comparing and simplifying fractions.], alt: "A mathematical equation questions whether the fraction 2/4 is equivalent to 1/2, illustrating a common concept in comparing and simplifying fractions.", caption: none)], [],
  [], [#figure(figph[A mathematical expression showing 1/2 equals 1/2, accompanied by a checkmark, likely indicating its correctness or validation.], alt: "A mathematical expression showing 1/2 equals 1/2, accompanied by a checkmark, likely indicating its correctness or validation.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the fraction 4 y minus 7 over 3 equals the fraction y over 6")[$frac(4 y − 7, 3) = frac(y, 6)$].

#solutionbox[
#math.equation(block: true, alt: "y equals 2")[$y = 2$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the fraction −2 z minus 5 over 4 equals the fraction z over 8")[$frac(−2 z − 5, 4) = frac(z, 8)$].

#solutionbox[
#math.equation(block: true, alt: "z equals −2")[$z = −2$]
]
]

#examplebox("Example 7")[][
Solve: #math.equation(block: false, alt: "the fraction a over 6 plus 2 equals the fraction a over 4 plus 3")[$frac(a, 6) + 2 = frac(a, 4) + 3$].

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[A mathematical equation is displayed, showing 'd/6 + 2 = d/4 + 3' in a horizontal layout against a white background.], alt: "A mathematical equation is displayed, showing 'd/6 + 2 = d/4 + 3' in a horizontal layout against a white background.", caption: none)], [],
  [Multiply by the LCD, 12.], [#figure(figph[A mathematical equation shows 12 multiplied by the sum of 'a' over 6 plus 2, equaling 12 multiplied by the sum of 'a' over 4 plus 3. The number 12 is highlighted in red on both sides.], alt: "A mathematical equation shows 12 multiplied by the sum of 'a' over 6 plus 2, equaling 12 multiplied by the sum of 'a' over 4 plus 3. The number 12 is highlighted in red on both sides.", caption: none)], [],
  [Distribute.], [#figure(figph[The mathematical equation 12 \* (d/6) + 12 \* 2 = 12 \* (d/4) + 12 \* 3 is displayed, demonstrating an algebraic problem with fractions and a variable 'd'.], alt: "The mathematical equation 12 * (d/6) + 12 * 2 = 12 * (d/4) + 12 * 3 is displayed, demonstrating an algebraic problem with fractions and a variable 'd'.", caption: none)], [],
  [Simplify.], [#figure(figph[The image shows the algebraic equation 2a + 24 = 3a + 36.], alt: "The image shows the algebraic equation 2a + 24 = 3a + 36.", caption: none)], [],
  [Collect the variables to the right.], [#figure(figph[A mathematical equation is displayed: 2a - 2a + 24 = 3a - 2a + 36. The terms '-2a' on both sides of the equation are highlighted in red.], alt: "A mathematical equation is displayed: 2a - 2a + 24 = 3a - 2a + 36. The terms '-2a' on both sides of the equation are highlighted in red.", caption: none)], [],
  [Simplify.], [#figure(figph[A mathematical equation shows '24 = q + 36' centered on a white background.], alt: "A mathematical equation shows '24 = q + 36' centered on a white background.", caption: none)], [],
  [Collect the constants to the left.], [#figure(figph[A mathematical equation shows 24 minus 36 equals 'a' plus 36 minus 36, illustrating the subtraction of 36 from both sides to isolate 'a'.], alt: "A mathematical equation shows 24 minus 36 equals 'a' plus 36 minus 36, illustrating the subtraction of 36 from both sides to isolate 'a'.", caption: none)], [],
  [Simplify.], [#figure(figph[The image displays a mathematical equation, 'a = 12,' written in a black font against a plain white background.], alt: "The image displays a mathematical equation, 'a = 12,' written in a black font against a plain white background.", caption: none)], [],
  [Check:], [#figure(figph[A mathematical equation is displayed, reading 'd over 6 plus 2 equals d over 4 plus 3' on a white background.], alt: "A mathematical equation is displayed, reading 'd over 6 plus 2 equals d over 4 plus 3' on a white background.", caption: none)], [],
  [Let #math.equation(block: false, alt: "a equals −12")[$a = −12$].], [#figure(figph[A mathematical equation is displayed, reading -12/6 + 2 =? -12/4 + 3, with a question mark placed above the equality sign, indicating a query about whether the two sides are equal.], alt: "A mathematical equation is displayed, reading -12/6 + 2 =? -12/4 + 3, with a question mark placed above the equality sign, indicating a query about whether the two sides are equal.", caption: none)], [],
  [], [#figure(figph[A mathematical equation: -2 + 2 =? -3 + 3. The equation questions if zero equals zero.], alt: "A mathematical equation: -2 + 2 =? -3 + 3. The equation questions if zero equals zero.", caption: none)], [],
  [], [#figure(figph[A mathematical expression showing '0 = 0V' written in a bold, dark font against a white background, appearing vertically oriented.], alt: "A mathematical expression showing '0 = 0V' written in a bold, dark font against a white background, appearing vertically oriented.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the fraction b over 10 plus 2 equals the fraction b over 4 plus 5")[$frac(b, 10) + 2 = frac(b, 4) + 5$].

#solutionbox[
#math.equation(block: true, alt: "b equals −20")[$b = −20$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the fraction c over 6 plus 3 equals the fraction c over 3 plus 4")[$frac(c, 6) + 3 = frac(c, 3) + 4$].

#solutionbox[
#math.equation(block: true, alt: "c equals −6")[$c = −6$]
]
]

#examplebox("Example 8")[][
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: 2, align: left, inset: 6pt,
  [], [#figure(figph[A mathematical equation is displayed on a white background: (4q + 3) divided by 2, plus 6, equals (3q + 5) divided by 4.], alt: "A mathematical equation is displayed on a white background: (4q + 3) divided by 2, plus 6, equals (3q + 5) divided by 4.", caption: none)],
  [Multiply by the LCD, 4.], [#figure(figph[Four times the quantity of four-halves q plus three-halves, plus six, equals four times the quantity of three-fourths of q plus five-fourths.], alt: "Four times the quantity of four-halves q plus three-halves, plus six, equals four times the quantity of three-fourths of q plus five-fourths.", caption: none)],
  [Distribute.], [#figure(figph[A mathematical equation showing 4 multiplied by the fraction (4q+3)/2, added to the product of 4 and 6, which equals 4 multiplied by the fraction (3q+5)/4.], alt: "A mathematical equation showing 4 multiplied by the fraction (4q+3)/2, added to the product of 4 and 6, which equals 4 multiplied by the fraction (3q+5)/4.", caption: none)],
  [Simplify.], [#figure(figph[An algebraic equation is shown, displaying 2(4q + 3) + 24 = 3q + 5, which needs to be solved for the variable 'q'.], alt: "An algebraic equation is shown, displaying 2(4q + 3) + 24 = 3q + 5, which needs to be solved for the variable 'q'.", caption: none)

 #linebreak() #figure(figph[A mathematical equation is displayed, showing '8q + 6 + 24 = 3q + 5' on a white background.], alt: "A mathematical equation is displayed, showing '8q + 6 + 24 = 3q + 5' on a white background.", caption: none)

 #linebreak() #figure(figph[A mathematical equation is displayed against a white background, reading '8q + 30 = 3q + 5'.], alt: "A mathematical equation is displayed against a white background, reading '8q + 30 = 3q + 5'.", caption: none)],
  [Collect the variables to the left.], [#figure(figph[An algebraic equation showing '8q - 3q + 30 = 3q - 3q + 5' with some terms in red, indicating a step in solving the equation where like terms might be identified or combined.], alt: "An algebraic equation showing '8q - 3q + 30 = 3q - 3q + 5' with some terms in red, indicating a step in solving the equation where like terms might be identified or combined.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation is displayed, showing '5q + 30 = 5' in the center of a white background. The numbers and symbols are in a dark gray font.], alt: "A mathematical equation is displayed, showing '5q + 30 = 5' in the center of a white background. The numbers and symbols are in a dark gray font.", caption: none)],
  [Collect the constants to the right.], [#figure(figph[A mathematical equation is displayed: 5q + 30 - 30 = 5 - 30, with the second '30' on the left and the '30' on the right highlighted in red.], alt: "A mathematical equation is displayed: 5q + 30 - 30 = 5 - 30, with the second '30' on the left and the '30' on the right highlighted in red.", caption: none)],
  [Simplify.], [#figure(figph[The image displays a simple algebraic equation in black text on a white background, which reads '5q = -25'.], alt: "The image displays a simple algebraic equation in black text on a white background, which reads '5q = -25'.", caption: none)],
  [Divide by 5.], [#figure(figph[The image shows the equation 5q/5 = -25/5, demonstrating the step of dividing both sides of an algebraic equation by 5 to solve for 'q'.], alt: "The image shows the equation 5q/5 = -25/5, demonstrating the step of dividing both sides of an algebraic equation by 5 to solve for 'q'.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression on a white background, showing 'q = -5' in black text.], alt: "A mathematical expression on a white background, showing 'q = -5' in black text.", caption: none)],
  [Check:], [#figure(figph[A mathematical equation is shown: (4q + 3) / 2 + 6 = (3q + 5) / 4.], alt: "A mathematical equation is shown: (4q + 3) / 2 + 6 = (3q + 5) / 4.", caption: none)],
  [Let #math.equation(block: false, alt: "q equals −5")[$q = −5$].], [#figure(figph[A mathematical equation featuring a question mark between two algebraic expressions, requiring determination of the relationship: (4(-5)+3)/2+6 ? (3(-5)+5)/4.], alt: "A mathematical equation featuring a question mark between two algebraic expressions, requiring determination of the relationship: (4(-5)+3)/2+6 ? (3(-5)+5)/4.", 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")[$s = −8$]
]
]


=== Solve Equations with Decimal Coefficients

Some equations have decimals in them. This kind of equation will 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: "0.3 equals the fraction 3 over 10")[$0.3 = frac(3, 10)$] and #math.equation(block: false, alt: "0.17 equals the fraction 17 over 100")[$0.17 = frac(17, 100)$]. 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 least common denominator.

#examplebox("Example 9")[][
Solve: #math.equation(block: false, alt: "0.06 x plus 0.02 equals 0.25 x minus 1.5")[$0.06 x + 0.02 = 0.25 x − 1.5$].

#solutionbox[
Look at the decimals and think of the equivalent fractions.

#math.equation(block: true, alt: "0.06 equals the fraction 6 over 100 0.02 equals the fraction 2 over 100 0.25 equals the fraction 25 over 100 1.5 equals 1 the fraction 5 over 10")[$0.06 = frac(6, 100) #h(2em) 0.02 = frac(2, 100) #h(2em) 0.25 = frac(25, 100) #h(2em) 1.5 = 1 frac(5, 10)$]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,
  table.header([], [#figure(figph[A linear equation is displayed: 0.06x + 0.02 = 0.25x - 1.5. It involves variables and constants with decimal values, requiring algebraic steps to solve for x.], alt: "A linear equation is displayed: 0.06x + 0.02 = 0.25x - 1.5. It involves variables and constants with decimal values, requiring algebraic steps to solve for x.", caption: none)]),
  [Multiply both sides by 100.], [#figure(figph[A mathematical equation, 100(0.06x + 0.02) = 100(0.25x - 1.5), where 100 is multiplied by two different linear expressions, displayed with the number 100 highlighted in red.], alt: "A mathematical equation, 100(0.06x + 0.02) = 100(0.25x - 1.5), where 100 is multiplied by two different linear expressions, displayed with the number 100 highlighted in red.", caption: none)],
  [Distribute.], [#figure(figph[A mathematical equation is displayed: 100(0.06x) + 100(0.02) = 100(0.25x) - 100(1.5). The expression involves constants, variables, and decimal numbers within parentheses, multiplied by 100.], alt: "A mathematical equation is displayed: 100(0.06x) + 100(0.02) = 100(0.25x) - 100(1.5). The expression involves constants, variables, and decimal numbers within parentheses, multiplied by 100.", caption: none)],
  [Multiply, and now we have no more decimals.], [#figure(figph[A mathematical equation is displayed with a white background showing '6x + 2 = 25x - 150' in a black font.], alt: "A mathematical equation is displayed with a white background showing '6x + 2 = 25x - 150' in a black font.", caption: none)],
  [Collect the variables to the right.], [#figure(figph[A mathematical equation is displayed, showing '6x - 6x + 2 = 25x - 6x - 150'. The terms '-6x' appear in red on both sides of the equation.], alt: "A mathematical equation is displayed, showing '6x - 6x + 2 = 25x - 6x - 150'. The terms '-6x' appear in red on both sides of the equation.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation is displayed against a white background, reading '2 = 19x - 150' in black text.], alt: "A mathematical equation is displayed against a white background, reading '2 = 19x - 150' in black text.", caption: none)],
  [Collect the constants to the left.], [#figure(figph[The equation 2 + 150 = 19x - 150 + 150, illustrating a step in solving for x by adding 150 to both sides.], alt: "The equation 2 + 150 = 19x - 150 + 150, illustrating a step in solving for x by adding 150 to both sides.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation '152 = 19x' is displayed against a white background, representing a linear equation to be solved for the variable 'x'.], alt: "A mathematical equation '152 = 19x' is displayed against a white background, representing a linear equation to be solved for the variable 'x'.", caption: none)],
  [Divide by 19.], [#figure(figph[A mathematical equation displays '152/19 = 19x/19'. The number '19' in the denominator is highlighted in red on both sides, illustrating the division of both terms by 19 to solve for x.], alt: "A mathematical equation displays '152/19 = 19x/19'. The number '19' in the denominator is highlighted in red on both sides, illustrating the division of both terms by 19 to solve for x.", caption: none)],
  [Simplify.], [#figure(figph[The number 8 is equal to the variable x, as shown in this simple algebraic equation.], alt: "The number 8 is equal to the variable x, as shown in this simple algebraic equation.", caption: none)],
  [Check:], [Let #math.equation(block: false, alt: "x equals 8")[$x = 8$]. #linebreak() #figure(figph[The image displays the step-by-step solution and verification of the equation 0.06(8) + 0.02 = 0.25(8) - 1.5, clearly showing how both sides simplify to 0.50.], alt: "The image displays the step-by-step solution and verification of the equation 0.06(8) + 0.02 = 0.25(8) - 1.5, clearly showing how both sides simplify to 0.50.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "0.14 h plus 0.12 equals 0.35 h minus 2.4")[$0.14 h + 0.12 = 0.35 h − 2.4$].

#solutionbox[
#math.equation(block: true, alt: "h equals 12")[$h = 12$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "0.65 k minus 0.1 equals 0.4 k minus 0.35")[$0.65 k − 0.1 = 0.4 k − 0.35$].

#solutionbox[
#math.equation(block: true, alt: "k equals −1")[$k = −1$]
]
]

The next example uses an equation that is typical of the money applications in the next chapter. Notice that we distribute the decimal before we clear all the decimals.

#examplebox("Example 10")[][
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[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed on a white background: 0.25x + 0.05(x + 3) = 2.85. The text is black and clearly visible, representing a linear equation to be solved for the variable x.], alt: "A mathematical equation is displayed on a white background: 0.25x + 0.05(x + 3) = 2.85. The text is black and clearly visible, representing a linear equation to be solved for the variable x.", caption: none)]),
  [Distribute first.], [#figure(figph[A mathematical equation is displayed against a white background: 0.25x + 0.05x + 0.15 = 2.85. The text is rendered in a dark gray, sans-serif font.], alt: "A mathematical equation is displayed against a white background: 0.25x + 0.05x + 0.15 = 2.85. The text is rendered in a dark gray, sans-serif font.", caption: none)],
  [Combine like terms.], [#figure(figph[A close-up image displaying the mathematical equation: 0.30x + 0.15 = 2.85. The numbers and symbols are clearly visible against a white background.], alt: "A close-up image displaying the mathematical equation: 0.30x + 0.15 = 2.85. The numbers and symbols are clearly visible against a white background.", caption: none)],
  [To clear decimals, multiply by 100.], [#figure(figph[The image displays the mathematical equation 100(0.30x + 0.15) = 100(2.85), with the number 100 on both sides highlighted in red.], alt: "The image displays the mathematical equation 100(0.30x + 0.15) = 100(2.85), with the number 100 on both sides highlighted in red.", caption: none)],
  [Distribute.], [#figure(figph[A mathematical equation is displayed on a white background, which reads '30x + 15 = 285'.], alt: "A mathematical equation is displayed on a white background, which reads '30x + 15 = 285'.", caption: none)],
  [Subtract 15 from both sides.], [#figure(figph[A mathematical equation is displayed, showing '30x + 15 - 15 = 285 - 15'. The numbers '- 15' are highlighted in red on both sides of the equation.], alt: "A mathematical equation is displayed, showing '30x + 15 - 15 = 285 - 15'. The numbers '- 15' 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 '30x = 270' in dark gray text.], alt: "A mathematical equation is displayed on a white background, which reads '30x = 270' in dark gray text.", caption: none)],
  [Divide by 30.], [#figure(figph[A mathematical equation shows '30x divided by 30 equals 270 divided by 30.' The denominator '30' on both sides is highlighted in red, indicating division to solve for x.], alt: "A mathematical equation shows '30x divided by 30 equals 270 divided by 30.' The denominator '30' on both sides is highlighted in red, indicating division to solve for x.", caption: none)],
  [Simplify.], [#figure(figph[The mathematical equation 'x = 9' is displayed against a white background.], alt: "The mathematical equation 'x = 9' is displayed against a white background.", 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[Strategy to Solve an Equation with Fraction Coefficients]
+ Find the least common denominator of all the fractions in the equation.
+ Multiply both sides of the equation by that LCD. This clears the fractions.
+ Solve using the General Strategy for Solving Linear Equations.


==== Practice Makes Perfect

#strong[Solve Equations with Fraction 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")[$frac(1, 4) x − frac(1, 2) = − frac(3, 4)$]

#math.equation(block: true, alt: "the fraction 3 over 4 x minus the fraction 1 over 2 equals the fraction 1 over 4")[$frac(3, 4) x − frac(1, 2) = frac(1, 4)$]

#solutionbox[
#math.equation(block: true, alt: "x equals 1")[$x = 1$]
]

#math.equation(block: true, alt: "the fraction 5 over 6 y minus the fraction 2 over 3 equals minus the fraction 3 over 2")[$frac(5, 6) y − frac(2, 3) = − frac(3, 2)$]

#math.equation(block: true, alt: "the fraction 5 over 6 y minus the fraction 1 over 3 equals minus the fraction 7 over 6")[$frac(5, 6) y − frac(1, 3) = − frac(7, 6)$]

#solutionbox[
#math.equation(block: true, alt: "y equals −1")[$y = −1$]
]

#math.equation(block: true, alt: "the fraction 1 over 2 a plus the fraction 3 over 8 equals the fraction 3 over 4")[$frac(1, 2) a + frac(3, 8) = frac(3, 4)$]

#math.equation(block: true, alt: "the fraction 5 over 8 b plus the fraction 1 over 2 equals minus the fraction 3 over 4")[$frac(5, 8) b + frac(1, 2) = − frac(3, 4)$]

#solutionbox[
#math.equation(block: true, alt: "b equals −2")[$b = −2$]
]

#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")[$2 = frac(1, 3) x − frac(1, 2) x + frac(2, 3) x$]

#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")[$2 = frac(3, 5) x − frac(1, 3) x + frac(2, 5) x$]

#solutionbox[
#math.equation(block: true, alt: "x equals 3")[$x = 3$]
]

#math.equation(block: true, alt: "the fraction 1 over 4 m minus the fraction 4 over 5 m plus the fraction 1 over 2 m equals −1")[$frac(1, 4) m − frac(4, 5) m + frac(1, 2) m = −1$]

#math.equation(block: true, alt: "the fraction 5 over 6 n minus the fraction 1 over 4 n minus the fraction 1 over 2 n equals −2")[$frac(5, 6) n − frac(1, 4) n − frac(1, 2) n = −2$]

#solutionbox[
#math.equation(block: true, alt: "n equals −24")[$n = −24$]
]

#math.equation(block: true, alt: "x plus the fraction 1 over 2 equals the fraction 2 over 3 x minus the fraction 1 over 2")[$x + frac(1, 2) = frac(2, 3) x − frac(1, 2)$]

#math.equation(block: true, alt: "x plus the fraction 3 over 4 equals the fraction 1 over 2 x minus the fraction 5 over 4")[$x + frac(3, 4) = frac(1, 2) x − frac(5, 4)$]

#solutionbox[
#math.equation(block: true, alt: "x equals −4")[$x = −4$]
]

#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")[$frac(1, 3) w + frac(5, 4) = w − frac(1, 4)$]

#math.equation(block: true, alt: "the fraction 3 over 2 z plus the fraction 1 over 3 equals z minus the fraction 2 over 3")[$frac(3, 2) z + frac(1, 3) = z − frac(2, 3)$]

#solutionbox[
#math.equation(block: true, alt: "z equals −2")[$z = −2$]
]

#math.equation(block: true, alt: "the fraction 1 over 2 x minus the fraction 1 over 4 equals the fraction 1 over 12 x plus the fraction 1 over 6")[$frac(1, 2) x − frac(1, 4) = frac(1, 12) x + frac(1, 6)$]

#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")[$frac(1, 2) a − frac(1, 4) = frac(1, 6) a + frac(1, 12)$]

#solutionbox[
#math.equation(block: true, alt: "a equals 1")[$a = 1$]
]

#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")[$frac(1, 3) b + frac(1, 5) = frac(2, 5) b − frac(3, 5)$]

#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")[$frac(1, 3) x + frac(2, 5) = frac(1, 5) x − frac(2, 5)$]

#solutionbox[
#math.equation(block: true, alt: "x equals −6")[$x = −6$]
]

#math.equation(block: true, alt: "1 equals the fraction 1 over 6 open parenthesis 12 x minus 6 close parenthesis")[$1 = frac(1, 6) ( 12 x − 6 )$]

#math.equation(block: true, alt: "1 equals the fraction 1 over 5 open parenthesis 15 x minus 10 close parenthesis")[$1 = frac(1, 5) ( 15 x − 10 )$]

#solutionbox[
#math.equation(block: true, alt: "x equals 1")[$x = 1$]
]

#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")[$frac(1, 4) ( p − 7 ) = frac(1, 3) ( p + 5 )$]

#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")[$frac(1, 5) ( q + 3 ) = frac(1, 2) ( q − 3 )$]

#solutionbox[
#math.equation(block: true, alt: "q equals 7")[$q = 7$]
]

#math.equation(block: true, alt: "the fraction 1 over 2 open parenthesis x plus 4 close parenthesis equals the fraction 3 over 4")[$frac(1, 2) ( x + 4 ) = frac(3, 4)$]

#math.equation(block: true, alt: "the fraction 1 over 3 open parenthesis x plus 5 close parenthesis equals the fraction 5 over 6")[$frac(1, 3) ( x + 5 ) = frac(5, 6)$]

#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 5 q minus 8 over 5 equals the fraction 2 q over 10")[$frac(5 q − 8, 5) = frac(2 q, 10)$]

#math.equation(block: true, alt: "the fraction 4 m plus 2 over 6 equals the fraction m over 3")[$frac(4 m + 2, 6) = frac(m, 3)$]

#solutionbox[
#math.equation(block: true, alt: "m equals −1")[$m = −1$]
]

#math.equation(block: true, alt: "the fraction 4 n plus 8 over 4 equals the fraction n over 3")[$frac(4 n + 8, 4) = frac(n, 3)$]

#math.equation(block: true, alt: "the fraction 3 p plus 6 over 3 equals the fraction p over 2")[$frac(3 p + 6, 3) = frac(p, 2)$]

#solutionbox[
#math.equation(block: true, alt: "p equals −4")[$p = −4$]
]

#math.equation(block: true, alt: "the fraction u over 3 minus 4 equals the fraction u over 2 minus 3")[$frac(u, 3) − 4 = frac(u, 2) − 3$]

#math.equation(block: true, alt: "the fraction v over 10 plus 1 equals the fraction v over 4 minus 2")[$frac(v, 10) + 1 = frac(v, 4) − 2$]

#solutionbox[
#math.equation(block: true, alt: "v equals 20")[$v = 20$]
]

#math.equation(block: true, alt: "the fraction c over 15 plus 1 equals the fraction c over 10 minus 1")[$frac(c, 15) + 1 = frac(c, 10) − 1$]

#math.equation(block: true, alt: "the fraction d over 6 plus 3 equals the fraction d over 8 plus 2")[$frac(d, 6) + 3 = frac(d, 8) + 2$]

#solutionbox[
#math.equation(block: true, alt: "d equals −24")[$d = −24$]
]

#math.equation(block: true, alt: "the fraction 3 x plus 4 over 2 plus 1 equals the fraction 5 x plus 10 over 8")[$frac(3 x + 4, 2) + 1 = frac(5 x + 10, 8)$]

#math.equation(block: true, alt: "the fraction 10 y minus 2 over 3 plus 3 equals the fraction 10 y plus 1 over 9")[$frac(10 y − 2, 3) + 3 = frac(10 y + 1, 9)$]

#solutionbox[
#math.equation(block: true, alt: "y equals −1")[$y = −1$]
]

#math.equation(block: true, alt: "the fraction 7 u minus 1 over 4 minus 1 equals the fraction 4 u plus 8 over 5")[$frac(7 u − 1, 4) − 1 = frac(4 u + 8, 5)$]

#math.equation(block: true, alt: "the fraction 3 v minus 6 over 2 plus 5 equals the fraction 11 v minus 4 over 5")[$frac(3 v − 6, 2) + 5 = frac(11 v − 4, 5)$]

#solutionbox[
#math.equation(block: true, alt: "v equals 4")[$v = 4$]
]

#strong[Solve Equations with Decimal Coefficients]

In the following exercises, solve each equation with decimal coefficients.

#math.equation(block: true, alt: "0.6 y plus 3 equals 9")[$0.6 y + 3 = 9$]

#math.equation(block: true, alt: "0.4 y minus 4 equals 2")[$0.4 y − 4 = 2$]

#solutionbox[
#math.equation(block: true, alt: "y equals 15")[$y = 15$]
]

#math.equation(block: true, alt: "3.6 j minus 2 equals 5.2")[$3.6 j − 2 = 5.2$]

#math.equation(block: true, alt: "2.1 k plus 3 equals 7.2")[$2.1 k + 3 = 7.2$]

#solutionbox[
#math.equation(block: true, alt: "k equals 2")[$k = 2$]
]

#math.equation(block: true, alt: "0.4 x plus 0.6 equals 0.5 x minus 1.2")[$0.4 x + 0.6 = 0.5 x − 1.2$]

#math.equation(block: true, alt: "0.7 x plus 0.4 equals 0.6 x plus 2.4")[$0.7 x + 0.4 = 0.6 x + 2.4$]

#solutionbox[
#math.equation(block: true, alt: "x equals 20")[$x = 20$]
]

#math.equation(block: true, alt: "0.23 x plus 1.47 equals 0.37 x minus 1.05")[$0.23 x + 1.47 = 0.37 x − 1.05$]

#math.equation(block: true, alt: "0.48 x plus 1.56 equals 0.58 x minus 0.64")[$0.48 x + 1.56 = 0.58 x − 0.64$]

#solutionbox[
#math.equation(block: true, alt: "x equals 22")[$x = 22$]
]

#math.equation(block: true, alt: "0.9 x minus 1.25 equals 0.75 x plus 1.75")[$0.9 x − 1.25 = 0.75 x + 1.75$]

#math.equation(block: true, alt: "1.2 x minus 0.91 equals 0.8 x plus 2.29")[$1.2 x − 0.91 = 0.8 x + 2.29$]

#solutionbox[
#math.equation(block: true, alt: "x equals 8")[$x = 8$]
]

#math.equation(block: true, alt: "0.05 n plus 0.10 open parenthesis n plus 8 close parenthesis equals 2.15")[$0.05 n + 0.10 ( n + 8 ) = 2.15$]

#math.equation(block: true, alt: "0.05 n plus 0.10 open parenthesis n plus 7 close parenthesis equals 3.55")[$0.05 n + 0.10 ( n + 7 ) = 3.55$]

#solutionbox[
#math.equation(block: true, alt: "n equals 19")[$n = 19$]
]

#math.equation(block: true, alt: "0.10 d plus 0.25 open parenthesis d plus 5 close parenthesis equals 4.05")[$0.10 d + 0.25 ( d + 5 ) = 4.05$]

#math.equation(block: true, alt: "0.10 d plus 0.25 open parenthesis d plus 7 close parenthesis equals 5.25")[$0.10 d + 0.25 ( d + 7 ) = 5.25$]

#solutionbox[
#math.equation(block: true, alt: "d equals 10")[$d = 10$]
]

#math.equation(block: true, alt: "0.05 open parenthesis q minus 5 close parenthesis plus 0.25 q equals 3.05")[$0.05 ( q − 5 ) + 0.25 q = 3.05$]

#math.equation(block: true, alt: "0.05 open parenthesis q minus 8 close parenthesis plus 0.25 q equals 4.10")[$0.05 ( q − 8 ) + 0.25 q = 4.10$]

#solutionbox[
#math.equation(block: true, alt: "q equals 15")[$q = 15$]
]


==== Everyday Math

#strong[Coins] Taylor has \$2.00 in dimes and pennies. The number of pennies is 2 more than the number of dimes. Solve the equation #math.equation(block: false, alt: "0.10 d plus 0.01 open parenthesis d plus 2 close parenthesis equals 2")[$0.10 d + 0.01 ( d + 2 ) = 2$] for #math.equation(block: false, alt: "d")[$d$], the number of dimes.

#strong[Stamps] Paula bought \$22.82 worth of 49-cent stamps and 21-cent stamps. The number of 21-cent stamps was 8 less than the number of 49-cent stamps. Solve the equation #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.

#solutionbox[
#math.equation(block: true, alt: "s equals 35")[$s = 35$]
]


==== Writing Exercises

Explain how you find the least common denominator of #math.equation(block: false, alt: "the fraction 3 over 8")[$frac(3, 8)$], #math.equation(block: false, alt: "the fraction 1 over 6")[$frac(1, 6)$], and #math.equation(block: false, alt: "the fraction 2 over 3")[$frac(2, 3)$].

If an equation has several fractions, how does multiplying both sides by the LCD make it easier to solve?

#solutionbox[
Answers will vary.
]

If an equation has fractions only on one side, why do you have to multiply both sides of the equation by the LCD?

In the equation #math.equation(block: false, alt: "0.35 x plus 2.1 equals 3.85")[$0.35 x + 2.1 = 3.85$] what is the LCD? How do you know?

#solutionbox[
100. Justifications will vary.
]


==== Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives 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 with fraction coefficients,” and “solve equations with decimal coefficients.” 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 with fraction coefficients,” and “solve equations with decimal coefficients.” The rest of the cells are blank.", caption: none)

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not?