#set document(title: "8.4 Solve Equations with Fraction or Decimal Coefficients", author: "OpenStax / XYZ Homework")
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== 8.4#h(0.6em)Solve Equations with Fraction or Decimal Coefficients

#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 and the fraction 1 over 4 .")[$frac(5, 6) #h(0.2em) "and" #h(0.2em) frac(1, 4) .$] #linebreak() 
If you missed this problem, review

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Multiply: #math.equation(block: false, alt: "4.78")[$4.78$] by #math.equation(block: false, alt: "100 .")[$100 .$] #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 .")[$frac(1, 8) #h(0.1em) x + frac(1, 2) = frac(1, 4) .$]

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed: 1/8x + 1/2 = 1/4. This is a linear equation with one variable, x, involving fractions.], alt: "A mathematical equation is displayed: 1/8x + 1/2 = 1/4. This is a linear equation with one variable, x, involving fractions.", 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[An algebraic equation: 1/8x + 1/2 - 1/2 = 1/4 - 1/2, with the -1/2 terms highlighted in red, likely for simplification.], alt: "An algebraic equation: 1/8x + 1/2 - 1/2 = 1/4 - 1/2, with the -1/2 terms highlighted in red, likely for simplification.", caption: none)],
  [Simplify the left side.], [#figure(figph[A mathematical equation is displayed, showing '1/8x = 1/4 - 1/2' in black text against a white background.], alt: "A mathematical equation is displayed, showing '1/8x = 1/4 - 1/2' in black text against a white background.", caption: none)],
  [Change the constants to equivalent fractions with the LCD.], [#figure(figph[A mathematical equation is displayed: 1/8x = 1/4 - 2/4. The equation involves fractions and a variable 'x'.], alt: "A mathematical equation is displayed: 1/8x = 1/4 - 2/4. The equation involves fractions and a variable 'x'.", caption: none)],
  [Subtract.], [#figure(figph[The image displays the algebraic equation '1/8x = -1/4' centered on a white background.], alt: "The image displays the algebraic equation '1/8x = -1/4' centered on a white background.", caption: none)],
  [Multiply both sides by the reciprocal of #math.equation(block: false, alt: "the fraction 1 over 8")[$frac(1, 8)$].], [#figure(figph[The equation (8/1)\*(1/8)x = (8/1)(-1/4) illustrates multiplying both sides by 8/1 (highlighted in red) to solve for the variable x.], alt: "The equation (8/1)*(1/8)x = (8/1)(-1/4) illustrates multiplying both sides by 8/1 (highlighted in red) to solve for the variable x.", caption: none)],
  [Simplify.], [#figure(figph[The equation x = -2 is displayed, representing a vertical line on a coordinate plane or a simple algebraic solution for x.], alt: "The equation x = -2 is displayed, representing a vertical line on a coordinate plane or a simple algebraic solution for x.", caption: none)],
))

This method worked fine, but many students don’t 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 #strong[Multiplication Property of Equality] and multiply both sides of an equation by the least common denominator of #emph[all] the fractions in the equation. The result of this operation will be a new equation, equivalent to the first, but with no fractions. This process is called #emph[clearing the equation of fractions]. Let’s solve the same equation again, but this time use the method that clears the fractions.

#examplebox("Example 1")[][
Solve: #math.equation(block: false, alt: "the fraction 1 over 8 x plus the fraction 1 over 2 equals the fraction 1 over 4 .")[$frac(1, 8) #h(0.1em) x + frac(1, 2) = frac(1, 4) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Find the least common denominator of #emph[all] the fractions in the equation.], [#figure(figph[A mathematical equation is shown: 1/8x + 1/2 = 1/4, with 'LCD = 8' indicating the Least Common Denominator.], alt: "A mathematical equation is shown: 1/8x + 1/2 = 1/4, with 'LCD = 8' indicating the Least Common Denominator.", caption: none)]),
  [Multiply both sides of the equation by that LCD, 8. This clears the fractions.], [#figure(figph[A mathematical equation shows 8 multiplied by the sum of 1/8x and 1/2, which equals 8 multiplied by 1/4. This is a step in solving a linear equation, likely to clear denominators.], alt: "A mathematical equation shows 8 multiplied by the sum of 1/8x and 1/2, which equals 8 multiplied by 1/4. This is a step in solving a linear equation, likely to clear denominators.", caption: none)],
  [Use the Distributive Property.], [#figure(figph[An algebraic equation is shown, displaying 8 multiplied by 1/8x, plus 8 multiplied by 1/2, equals 8 multiplied by 1/4.], alt: "An algebraic equation is shown, displaying 8 multiplied by 1/8x, plus 8 multiplied by 1/2, equals 8 multiplied by 1/4.", caption: none)],
  [Simplify — and notice, no more fractions!], [#figure(figph[A mathematical equation is displayed, showing 'x + 4 = 2' in black text against a white background.], alt: "A mathematical equation is displayed, showing 'x + 4 = 2' in black text against a white background.", caption: none)],
  [Solve using the General Strategy for Solving Linear Equations.], [#figure(figph[An algebraic equation is shown where 4 is subtracted from both sides, with the subtracted 4s highlighted in red. The equation is x + 4 - 4 = 2 - 4.], alt: "An algebraic equation is shown where 4 is subtracted from both sides, with the subtracted 4s highlighted in red. The equation is x + 4 - 4 = 2 - 4.", caption: none)],
  [Simplify.], [#figure(figph[The image displays a simple mathematical equation, 'x = -2', written in black text on a white background. The equation indicates that the variable 'x' is equal to negative two.], alt: "The image displays a simple mathematical equation, 'x = -2', written in black text on a white background. The equation indicates that the variable 'x' is equal to negative two.", caption: none)],
  [Check: Let #math.equation(block: false, alt: "x equals −2")[$x = −2$] #linebreak() #figure(figph[Algebraic steps verifying that x=-2 satisfies the equation (1/8)x + 1/2 = 1/4, leading to the equality 1/4 = 1/4.], alt: "Algebraic steps verifying that x=-2 satisfies the equation (1/8)x + 1/2 = 1/4, leading to the equality 1/4 = 1/4.", 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 .")[$frac(1, 4) #h(0.1em) x + frac(1, 2) = frac(5, 8) .$]

#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 6 y minus the fraction 1 over 3 equals the fraction 1 over 6 .")[$frac(1, 6) #h(0.1em) y − frac(1, 3) = frac(1, 6) .$]

#solutionbox[
#emph[y] = 3
]
]

Notice in  that 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[Solve equations with fraction coefficients by clearing the fractions.]

+ 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: "7 equals the fraction 1 over 2 x plus the fraction 3 over 4 x minus the fraction 2 over 3 x .")[$7 = frac(1, 2) #h(0.1em) x + frac(3, 4) #h(0.1em) x − frac(2, 3) #h(0.1em) x .$]

#solutionbox[
We want to clear the fractions by multiplying both sides of the equation by the LCD of all the fractions in the equation. #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Find the least common denominator of #emph[all] the fractions in the equation.], [#figure(figph[A mathematical equation shows '7 = (1/2)x + (3/4)x - (2/3)x' followed by 'LCD = 12', indicating the least common denominator for the fractions in the expression.], alt: "A mathematical equation shows '7 = (1/2)x + (3/4)x - (2/3)x' followed by 'LCD = 12', indicating the least common denominator for the fractions in the expression.", caption: none)]),
  [Multiply both sides of the equation by 12.], [#figure(figph[A mathematical equation is displayed, reading '12(7) = 12 \* (1/2x + 3/4x - 2/3x)'. The numbers '12' on both sides of the equals sign are highlighted in red.], alt: "A mathematical equation is displayed, reading '12(7) = 12 * (1/2x + 3/4x - 2/3x)'. The numbers '12' on both sides of the equals sign are highlighted in red.", caption: none)],
  [Distribute.], [#figure(figph[A mathematical equation is displayed, showing 12 multiplied by 7 on the left side, equal to the sum and difference of three terms on the right side: 12 times 1/2x, plus 12 times 3/4x, minus 12 times 2/3x.], alt: "A mathematical equation is displayed, showing 12 multiplied by 7 on the left side, equal to the sum and difference of three terms on the right side: 12 times 1/2x, plus 12 times 3/4x, minus 12 times 2/3x.", caption: none)],
  [Simplify — and notice, no more fractions!], [#figure(figph[A mathematical equation is displayed, reading 84 = 6x + 9x - 8x. It's a linear equation with one variable 'x' on the right side, ready to be solved.], alt: "A mathematical equation is displayed, reading 84 = 6x + 9x - 8x. It's a linear equation with one variable 'x' on the right side, ready to be solved.", caption: none)],
  [Combine like terms.], [#figure(figph[A mathematical equation is displayed with the number 84 on the left side of an equals sign, and '7x' on the right side. The equation reads '84 = 7x'.], alt: "A mathematical equation is displayed with the number 84 on the left side of an equals sign, and '7x' on the right side. The equation reads '84 = 7x'.", caption: none)],
  [Divide by 7.], [#figure(figph[An algebraic equation showing 84/7 = 7x/7, with the common divisor 7 highlighted in red.], alt: "An algebraic equation showing 84/7 = 7x/7, with the common divisor 7 highlighted in red.", caption: none)],
  [Simplify.], [#figure(figph[A simple mathematical equation '12 = x' is displayed in black text on a plain white background, indicating that the variable 'x' is equal to the number 12.], alt: "A simple mathematical equation '12 = x' is displayed in black text on a plain white background, indicating that the variable 'x' is equal to the number 12.", caption: none)],
  [Check: Let #math.equation(block: false, alt: "x equals 12 .")[$x = 12 .$]], [],
  [#figure(figph[Mathematical solution showing the steps to solve the equation 7 = 1/2x + 3/4x - 2/3x, concluding with x=12 and a verification that 7 = 7. A clear example of algebraic problem-solving.], alt: "Mathematical solution showing the steps to solve the equation 7 = 1/2x + 3/4x - 2/3x, concluding with x=12 and a verification that 7 = 7. A clear example of algebraic problem-solving.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
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 .")[$6 = frac(1, 2) #h(0.1em) v + frac(2, 5) #h(0.1em) v − frac(3, 4) #h(0.1em) v .$]

#solutionbox[
#emph[v] = 40
]
]

#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 .")[$−1 = frac(1, 2) #h(0.1em) u + frac(1, 4) #h(0.1em) u − frac(2, 3) #h(0.1em) u .$]

#solutionbox[
#emph[u] = −12
]
]

In the next example, we’ll have variables and fractions on both sides of the equation.

#examplebox("Example 3")[][
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 .")[$x + frac(1, 3) = frac(1, 6) #h(0.1em) x − frac(1, 2) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Find the LCD of all the fractions in the equation.], [#figure(figph[A mathematical equation showing x + 1/3 = 1/6x - 1/2, with LCD = 6. This represents a linear equation with fractions where the least common denominator is 6.], alt: "A mathematical equation showing x + 1/3 = 1/6x - 1/2, with LCD = 6. This represents a linear equation with fractions where the least common denominator is 6.", caption: none)]),
  [Multiply both sides by the LCD.], [#figure(figph[A mathematical equation showing 6 multiplied by the sum of x and 1/3, which is equal to 6 multiplied by the difference of 1/6x and 1/2, with the number 6 highlighted in red.], alt: "A mathematical equation showing 6 multiplied by the sum of x and 1/3, which is equal to 6 multiplied by the difference of 1/6x and 1/2, with the number 6 highlighted in red.", caption: none)],
  [Distribute.], [#figure(figph[A mathematical equation showing the multiplication of fractions and a variable 'x': 6 \* x + 6 \* (1/3) = 6 \* (1/6) \* x - 6 \* (1/2).], alt: "A mathematical equation showing the multiplication of fractions and a variable 'x': 6 * x + 6 * (1/3) = 6 * (1/6) * x - 6 * (1/2).", caption: none)],
  [Simplify — no more fractions!], [#figure(figph[A mathematical equation is displayed, reading '6x + 2 = x - 3' in black text against a white background.], alt: "A mathematical equation is displayed, reading '6x + 2 = x - 3' in black text against a white background.", caption: none)],
  [Subtract #math.equation(block: false, alt: "x")[$x$] from both sides.], [#figure(figph[A mathematical equation is displayed on a white background: 6x - x + 2 = x - x - 3. Some instances of the variable 'x' are highlighted in red, specifically the second 'x' on the left and the second 'x' on the right side of the equation.], alt: "A mathematical equation is displayed on a white background: 6x - x + 2 = x - x - 3. Some instances of the variable 'x' are highlighted in red, specifically the second 'x' on the left and the second 'x' on the right side of the equation.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation is displayed on a white background. The equation reads '5x + 2 = -3' in black text, showing an algebraic expression set equal to a negative integer.], alt: "A mathematical equation is displayed on a white background. The equation reads '5x + 2 = -3' in black text, showing an algebraic expression set equal to a negative integer.", caption: none)],
  [Subtract 2 from both sides.], [#figure(figph[A mathematical equation shows '5x + 2 - 2 = -3 - 2', with the number '2' highlighted in red as it is subtracted from both sides, indicating a step in solving for 'x'.], alt: "A mathematical equation shows '5x + 2 - 2 = -3 - 2', with the number '2' highlighted in red as it is subtracted from both sides, indicating a step in solving for 'x'.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation is displayed on a white background, reading '5x = -5' in black font.], alt: "A mathematical equation is displayed on a white background, reading '5x = -5' in black font.", caption: none)],
  [Divide by 5.], [#figure(figph[A mathematical equation showing 5x divided by 5 equals -5 divided by 5. The number 5 in the denominator on both sides is highlighted in red, indicating a division step to solve for x.], alt: "A mathematical equation showing 5x divided by 5 equals -5 divided by 5. The number 5 in the denominator on both sides is highlighted in red, indicating a division step to solve for x.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation is displayed on a white background, reading 'x = -1' in a dark, serif-like font. The equation is centrally aligned, representing a simple algebraic solution.], alt: "A mathematical equation is displayed on a white background, reading 'x = -1' in a dark, serif-like font. The equation is centrally aligned, representing a simple algebraic solution.", caption: none)],
  [Check: Substitute #math.equation(block: false, alt: "x equals −1 .")[$x = −1 .$]], [],
  [#figure(figph[Verifying a linear equation's solution: Substituting x = -1 into x + 1/3 = 1/6x - 1/2 and simplifying both sides to confirm the equality, resulting in -2/3 = -2/3.], alt: "Verifying a linear equation's solution: Substituting x = -1 into x + 1/3 = 1/6x - 1/2 and simplifying both sides to confirm the equality, resulting in -2/3 = -2/3.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
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 .")[$a + frac(3, 4) = frac(3, 8) #h(0.1em) a − frac(1, 2) .$]

#solutionbox[
#emph[a] = −2
]
]

#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 .")[$c + frac(3, 4) = frac(1, 2) #h(0.1em) c − frac(1, 4) .$]

#solutionbox[
#emph[c] = −2
]
]

In , we’ll start by using the Distributive Property. This step will clear the fractions right away!

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

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed, showing '1 = (1/2)(4x + 2)'. The equation is presented in black text against a white background.], alt: "A mathematical equation is displayed, showing '1 = (1/2)(4x + 2)'. The equation is presented in black text against a white background.", caption: none)]),
  [Distribute.], [#figure(figph[An algebraic equation is shown: 1 = (1/2) \* 4x + (1/2) \* 2.], alt: "An algebraic equation is shown: 1 = (1/2) * 4x + (1/2) * 2.", caption: none)],
  [Simplify. Now there are no fractions to clear!], [#figure(figph[A mathematical equation is displayed, showing '1 = 2x + 1'.], alt: "A mathematical equation is displayed, showing '1 = 2x + 1'.", caption: none)],
  [Subtract 1 from both sides.], [#figure(figph[An algebraic equation reads '1 - 1 = 2x + 1 - 1'. The '1's that are being subtracted on both sides of the equals sign are visually emphasized in red, suggesting cancellation for simplification.], alt: "An algebraic equation reads '1 - 1 = 2x + 1 - 1'. The '1's that are being subtracted on both sides of the equals sign are visually emphasized in red, suggesting cancellation for simplification.", caption: none)],
  [Simplify.], [#figure(figph[The image displays a simple algebraic equation, 0 = 2x, on a white background. This equation implies that the value of x must be zero.], alt: "The image displays a simple algebraic equation, 0 = 2x, on a white background. This equation implies that the value of x must be zero.", caption: none)],
  [Divide by 2.], [#figure(figph[The image displays a mathematical equation: 0/2 = 2x/2. The denominator '2' is highlighted in red on both the left and right sides of the equality.], alt: "The image displays a mathematical equation: 0/2 = 2x/2. The denominator '2' is highlighted in red on both the left and right sides of the equality.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation shows '0 = x' on a white background.], alt: "A mathematical equation shows '0 = x' on a white background.", caption: none)],
  [Check: Let #math.equation(block: false, alt: "x equals 0 .")[$x = 0 .$]], [],
  [#figure(figph[This image demonstrates the verification of the equation 1 = 1/2(4x + 2) by substituting x=0. The process shows that both sides of the equation simplify to 1, thus confirming the equality.], alt: "This image demonstrates the verification of the equation 1 = 1/2(4x + 2) by substituting x=0. The process shows that both sides of the equation simplify to 1, thus confirming the equality.", 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[
#emph[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[
#emph[q] = 2
]
]

Many times, there will still be fractions, even after distributing.

#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[An algebraic equation showing one-half times the quantity y minus 5 equals one-fourth times the quantity y minus 1. The equation is: 1/2(y-5) = 1/4(y-1).], alt: "An algebraic equation showing one-half times the quantity y minus 5 equals one-fourth times the quantity y minus 1. The equation is: 1/2(y-5) = 1/4(y-1).", caption: none)]),
  [Distribute.], [#figure(figph[A mathematical equation is displayed, reading: one-half times y minus one-half times five equals one-fourth times y minus one-fourth times one.], alt: "A mathematical equation is displayed, reading: one-half times y minus one-half times five equals one-fourth times y minus one-fourth times one.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation showing one-half y minus five-halves equals one-fourth y minus one-fourth.], alt: "A mathematical equation showing one-half y minus five-halves equals one-fourth y minus one-fourth.", caption: none)],
  [Multiply by the LCD, 4.], [#figure(figph[An algebraic equation showing 4 multiplied by the quantity one-half y minus five-halves, equals 4 multiplied by the quantity one-fourth y minus one-fourth.], alt: "An algebraic equation showing 4 multiplied by the quantity one-half y minus five-halves, equals 4 multiplied by the quantity one-fourth y minus one-fourth.", caption: none)],
  [Distribute.], [#figure(figph[A mathematical equation shown in black text on a white background. The equation is '4 multiplied by 1/2y minus 4 multiplied by 5/2 equals 4 multiplied by 1/4y minus 4 multiplied by 1/4'.], alt: "A mathematical equation shown in black text on a white background. The equation is '4 multiplied by 1/2y minus 4 multiplied by 5/2 equals 4 multiplied by 1/4y minus 4 multiplied by 1/4'.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation is displayed, reading '2y - 10 = y - 1' in black text against a plain white background.], alt: "A mathematical equation is displayed, reading '2y - 10 = y - 1' in black text against a plain white background.", caption: none)],
  [Collect the #math.equation(block: false, alt: "y")[$y$] terms to the left.], [#figure(figph[A mathematical equation is displayed, showing '2y - 10 - y = y - 1 - y'. The terms '-y' on both sides of the equation are highlighted in red, indicating they might be cancelled out.], alt: "A mathematical equation is displayed, showing '2y - 10 - y = y - 1 - y'. The terms '-y' on both sides of the equation are highlighted in red, indicating they might be cancelled out.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation is displayed, reading 'y - 10 = -1' in black text against a white background. It represents a simple algebraic problem.], alt: "A mathematical equation is displayed, reading 'y - 10 = -1' in black text against a white background. It represents a simple algebraic problem.", caption: none)],
  [Collect the constants to the right.], [#figure(figph[An algebraic equation showing the step of adding 10 to both sides to solve for y: y - 10 + 10 = -1 + 10.], alt: "An algebraic equation showing the step of adding 10 to both sides to solve for y: y - 10 + 10 = -1 + 10.", caption: none)],
  [Simplify.], [#figure(figph[The equation y=9 is displayed in black text on a white background.], alt: "The equation y=9 is displayed in black text on a white background.", caption: none)],
  [Check: Substitute #math.equation(block: false, alt: "9")[$9$] for #math.equation(block: false, alt: "y .")[$y .$]], [],
  [#figure(figph[Verification of the solution y=9 for the equation 1/2(y-5) = 1/4(y-1). The steps show substituting 9 for y, simplifying both sides to 2, thus confirming the equality.], alt: "Verification of the solution y=9 for the equation 1/2(y-5) = 1/4(y-1). The steps show substituting 9 for y, simplifying both sides to 2, thus confirming the equality.", caption: none)], [],
))
]
]

#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[
#emph[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[
#emph[m] = −1
]
]


=== Solve Equations with Decimal Coefficients

Some equations have decimals in them. This kind of equation will occur when we solve problems dealing with money and percent. But decimals are really another way to represent 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, when we have an equation with decimals, we can use the same process we used to clear fractions—multiply both sides of the equation by the #strong[least common denominator].

#examplebox("Example 6")[][
Solve: #math.equation(block: false, alt: "0.8 x minus 5 equals 7 .")[$0.8 x − 5 = 7 .$]

#solutionbox[
The only decimal in the equation is #math.equation(block: false, alt: "0.8 .")[$0.8 .$] Since #math.equation(block: false, alt: "0.8 equals the fraction 8 over 10 ,")[$0.8 = frac(8, 10) ,$] the LCD is #math.equation(block: false, alt: "10 .")[$10 .$] We can multiply both sides by #math.equation(block: false, alt: "10")[$10$] to clear the decimal.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed against a white background, reading '0.8x - 5 = 7'.], alt: "A mathematical equation is displayed against a white background, reading '0.8x - 5 = 7'.", caption: none)]),
  [Multiply both sides by the LCD.], [#figure(figph[The mathematical equation 10(0.8x - 5) = 10(7) is displayed, with the number 10 highlighted in red on both sides of the equality sign, indicating a common factor or a step in solving for x.], alt: "The mathematical equation 10(0.8x - 5) = 10(7) is displayed, with the number 10 highlighted in red on both sides of the equality sign, indicating a common factor or a step in solving for x.", caption: none)],
  [Distribute.], [#figure(figph[A mathematical equation is shown: 10(0.8x) - 10(5) = 10(7). The equation involves multiplication and subtraction, with the number 10 being a common factor on both sides of the equality.], alt: "A mathematical equation is shown: 10(0.8x) - 10(5) = 10(7). The equation involves multiplication and subtraction, with the number 10 being a common factor on both sides of the equality.", caption: none)],
  [Multiply, and notice, no more decimals!], [#figure(figph[A mathematical equation displays '8x - 50 = 70' in black text on a white background, representing a linear equation with one variable to be solved.], alt: "A mathematical equation displays '8x - 50 = 70' in black text on a white background, representing a linear equation with one variable to be solved.", caption: none)],
  [Add 50 to get all constants to the right.], [#figure(figph[The equation 8x - 50 + 50 = 70 + 50 demonstrates adding 50 to both sides to isolate the term with x in a linear equation.], alt: "The equation 8x - 50 + 50 = 70 + 50 demonstrates adding 50 to both sides to isolate the term with x in a linear equation.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation shows '8x = 120' in black text on a white background.], alt: "A mathematical equation shows '8x = 120' in black text on a white background.", caption: none)],
  [Divide both sides by 8.], [#figure(figph[The equation 8x/8 = 120/8, showing division by 8 on both sides. The number 8 in the denominator is highlighted in red.], alt: "The equation 8x/8 = 120/8, showing division by 8 on both sides. The number 8 in the denominator is highlighted in red.", caption: none)],
  [Simplify.], [#figure(figph[The mathematical equation 'x = 15' is displayed in a clean, minimalist style against a white background.], alt: "The mathematical equation 'x = 15' is displayed in a clean, minimalist style against a white background.", caption: none)],
  [Check: Let #math.equation(block: false, alt: "x equals 15 .")[$x = 15 .$]], [],
  [#figure(figph[A step-by-step mathematical solution demonstrates that 0.8(15) - 5 equals 7. The calculation shows 12 - 5 simplifies to 7, confirming the equality with a checkmark.], alt: "A step-by-step mathematical solution demonstrates that 0.8(15) - 5 equals 7. The calculation shows 12 - 5 simplifies to 7, confirming the equality with a checkmark.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "0.6 x minus 1 equals 11 .")[$0.6 x − 1 = 11 .$]

#solutionbox[
#emph[x] = 20
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "1.2 x minus 3 equals 9 .")[$1.2 x − 3 = 9 .$]

#solutionbox[
#emph[x] = 10
]
]

#examplebox("Example 7")[][
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")[$0.06 = frac(6, 100) , #h(1em) 0.02 = frac(2, 100) , #h(1em) 0.25 = frac(25, 100) , #h(1em) 1.5 = 1 frac(5, 10)$]

Notice, the LCD is #math.equation(block: false, alt: "100 .")[$100 .$]

By multiplying by the LCD we will clear the decimals. #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A linear equation is displayed as '0.06x + 0.02 = 0.25x - 1.5'.], alt: "A linear equation is displayed as '0.06x + 0.02 = 0.25x - 1.5'.", caption: none)]),
  [Multiply both sides by 100.], [#figure(figph[An algebraic equation showing 100 multiplied by the sum of 0.06x and 0.02, equaling 100 multiplied by the difference of 0.25x and 1.5. The number 100 is highlighted in red.], alt: "An algebraic equation showing 100 multiplied by the sum of 0.06x and 0.02, equaling 100 multiplied by the difference of 0.25x and 1.5. The number 100 is 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).], alt: "A mathematical equation is displayed: 100(0.06x) + 100(0.02) = 100(0.25x) - 100(1.5).", caption: none)],
  [Multiply, and now no more decimals.], [#figure(figph[A mathematical equation is displayed against a white background, reading 6x + 2 = 25x - 150.], alt: "A mathematical equation is displayed against a white background, reading 6x + 2 = 25x - 150.", caption: none)],
  [Collect the variables to the right.], [#figure(figph[An algebraic equation: 6x - 6x + 2 = 25x - 6x - 150, with specific '6x' terms highlighted in red for emphasis.], alt: "An algebraic equation: 6x - 6x + 2 = 25x - 6x - 150, with specific '6x' terms highlighted in red for emphasis.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation is displayed, showing '2 = 19x - 150'. The equation contains the numbers 2, 19, and 150, along with the variable 'x', an equals sign, and a minus sign.], alt: "A mathematical equation is displayed, showing '2 = 19x - 150'. The equation contains the numbers 2, 19, and 150, along with the variable 'x', an equals sign, and a minus sign.", caption: none)],
  [Collect the constants to the left.], [#figure(figph[A mathematical equation, 2 + 150 = 19x - 150 + 150, is displayed in black text with the numbers '+ 150' on both sides highlighted in red, indicating a step in solving for x.], alt: "A mathematical equation, 2 + 150 = 19x - 150 + 150, is displayed in black text with the numbers '+ 150' on both sides highlighted in red, indicating a step in solving for x.", caption: none)],
  [Simplify.], [#figure(figph[A simple algebraic equation is presented, showing '152 = 19x' against a white background.], alt: "A simple algebraic equation is presented, showing '152 = 19x' against a white background.", caption: none)],
  [Divide by 19.], [#figure(figph[Algebraic equation: 152/19 = 19x/19, demonstrating division by 19 to isolate x.], alt: "Algebraic equation: 152/19 = 19x/19, demonstrating division by 19 to isolate x.", caption: none)],
  [Simplify.], [#figure(figph[The image shows a mathematical equation on a white background, stating '8 = x', indicating that the value of the variable x is equal to 8.], alt: "The image shows a mathematical equation on a white background, stating '8 = x', indicating that the value of the variable x is equal to 8.", caption: none)],
  [Check: Let #math.equation(block: false, alt: "x equals 8 .")[$x = 8 .$]], [],
  [#figure(figph[A three-step solution to a mathematical equation, showing 0.06(8) + 0.02 = 0.25(8) - 1.5 simplifies to 0.50 = 0.50, verified with a checkmark.], alt: "A three-step solution to a mathematical equation, showing 0.06(8) + 0.02 = 0.25(8) - 1.5 simplifies to 0.50 = 0.50, verified with a checkmark.", 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[
#emph[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[
#emph[k] = −1
]
]

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

#examplebox("Example 8")[][
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[The image shows the algebraic equation 0.25x + 0.05(x + 3) = 2.85.], alt: "The image shows the algebraic equation 0.25x + 0.05(x + 3) = 2.85.", caption: none)]),
  [Distribute first.], [#figure(figph[A mathematical equation is presented, reading '0.25x + 0.05x + 0.15 = 2.85' against a white background.], alt: "A mathematical equation is presented, reading '0.25x + 0.05x + 0.15 = 2.85' against a white background.", caption: none)],
  [Combine like terms.], [#figure(figph[A mathematical equation is displayed, reading '0.30x + 0.15 = 2.85' against a white background.], alt: "A mathematical equation is displayed, reading '0.30x + 0.15 = 2.85' against a white background.", caption: none)],
  [To clear decimals, multiply by 100.], [#figure(figph[A mathematical equation shows 100 multiplied by the sum of 0.30x and 0.15 on the left side, which is equal to 100 multiplied by 2.85 on the right side. The number 100 is highlighted in red.], alt: "A mathematical equation shows 100 multiplied by the sum of 0.30x and 0.15 on the left side, which is equal to 100 multiplied by 2.85 on the right side. The number 100 is 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[The equation 30x + 15 - 15 = 285 - 15, illustrating a step in solving for x where 15 is subtracted from both sides, with the subtracted 15 shown in red.], alt: "The equation 30x + 15 - 15 = 285 - 15, illustrating a step in solving for x where 15 is subtracted from both sides, with the subtracted 15 shown in red.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation showing 30 multiplied by x equals 270.], alt: "A mathematical equation showing 30 multiplied by x equals 270.", caption: none)],
  [Divide by 30.], [#figure(figph[A mathematical equation shows '30x / 30 = 270 / 30' with the denominator '30' on both sides highlighted in red.], alt: "A mathematical equation shows '30x / 30 = 270 / 30' with the denominator '30' on both sides highlighted in red.", caption: none)],
  [Simplify.], [#figure(figph[A simple mathematical equation 'x = 9' is displayed on a white background.], alt: "A simple mathematical equation 'x = 9' is displayed on a white background.", caption: none)],
  [Check: Let #math.equation(block: false, alt: "x equals 9 .")[$x = 9 .$]], [],
  [#figure(figph[Mathematical steps showing the verification of the solution x=9 for the equation 0.25x + 0.05(x + 3) = 2.85, concluding with a checked equality 2.85 = 2.85.], alt: "Mathematical steps showing the verification of the solution x=9 for the equation 0.25x + 0.05(x + 3) = 2.85, concluding with a checked equality 2.85 = 2.85.", caption: none)], [],
))
]
]

#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[
#emph[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[
#emph[d] = 16
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[ACCESS ADDITIONAL ONLINE RESOURCES]

- #link("https://www.openstax.org/l/24FracVariTerm")[Solve an Equation with Fractions with Variable Terms on Both Sides]
- #link("https://www.openstax.org/l/24Ex1VariTerms")[Ex 1: Solve an Equation with Fractions with Variable Terms on Both Sides]
- #link("https://www.openstax.org/l/24Ex2VariTerms")[Ex 2: Solve an Equation with Fractions with Variable Terms on Both Sides]
- #link("https://www.openstax.org/l/24EquawithDec")[Solving Multiple Step Equations Involving Decimals]
- #link("https://www.openstax.org/l/24LinEquaDec")[Ex: Solve a Linear Equation With Decimals and Variables on Both Sides]
- #link("https://www.openstax.org/l/24DecParens")[Ex: Solve an Equation with Decimals and Parentheses]
]


=== Key Concepts

- #emph[Solve equations with fraction coefficients by clearing the fractions.]+ 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.


=== Section Exercises


==== Practice Makes Perfect

#strong[Solve equations with fraction coefficients]

In the following exercises, solve the equation by clearing the fractions.

#math.equation(block: true, alt: "the fraction 1 over 4 x minus the fraction 1 over 2 equals minus the fraction 3 over 4")[$frac(1, 4) #h(0.1em) x − frac(1, 2) = − frac(3, 4)$]

#solutionbox[
#emph[x] = −1
]

#math.equation(block: true, alt: "the fraction 3 over 4 x minus the fraction 1 over 2 equals the fraction 1 over 4")[$frac(3, 4) #h(0.1em) x − frac(1, 2) = frac(1, 4)$]

#math.equation(block: true, alt: "the fraction 5 over 6 y minus the fraction 2 over 3 equals minus the fraction 3 over 2")[$frac(5, 6) #h(0.1em) y − frac(2, 3) = − frac(3, 2)$]

#solutionbox[
#emph[y] = −1
]

#math.equation(block: true, alt: "the fraction 5 over 6 y minus the fraction 1 over 3 equals minus the fraction 7 over 6")[$frac(5, 6) #h(0.1em) y − frac(1, 3) = − frac(7, 6)$]

#math.equation(block: true, alt: "the fraction 1 over 2 a plus the fraction 3 over 8 equals the fraction 3 over 4")[$frac(1, 2) #h(0.1em) a + frac(3, 8) = frac(3, 4)$]

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

#math.equation(block: true, alt: "the fraction 5 over 8 b plus the fraction 1 over 2 equals minus the fraction 3 over 4")[$frac(5, 8) #h(0.1em) b + frac(1, 2) = − frac(3, 4)$]

#math.equation(block: true, alt: "2 equals the fraction 1 over 3 x minus the fraction 1 over 2 x plus the fraction 2 over 3 x")[$2 = frac(1, 3) #h(0.1em) x − frac(1, 2) #h(0.1em) x + frac(2, 3) #h(0.1em) x$]

#solutionbox[
#emph[x] = 4
]

#math.equation(block: true, alt: "2 equals the fraction 3 over 5 x minus the fraction 1 over 3 x plus the fraction 2 over 5 x")[$2 = frac(3, 5) #h(0.1em) x − frac(1, 3) #h(0.1em) x + frac(2, 5) #h(0.1em) x$]

#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")[$frac(1, 4) #h(0.1em) m − frac(4, 5) #h(0.1em) m + frac(1, 2) #h(0.1em) m = −1$]

#solutionbox[
#emph[m] = 20
]

#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")[$frac(5, 6) #h(0.1em) n − frac(1, 4) #h(0.1em) n − frac(1, 2) #h(0.1em) n = −2$]

#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")[$x + frac(1, 2) = frac(2, 3) #h(0.1em) x − frac(1, 2)$]

#solutionbox[
#emph[x] = −3
]

#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")[$x + frac(3, 4) = frac(1, 2) #h(0.1em) x − frac(5, 4)$]

#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")[$frac(1, 3) #h(0.1em) w + frac(5, 4) = w − frac(1, 4)$]

#solutionbox[
#math.equation(block: true, alt: "w equals the fraction 9 over 4")[$w = frac(9, 4)$]
]

#math.equation(block: true, alt: "the fraction 3 over 2 z plus the fraction 1 over 3 equals z minus the fraction 2 over 3")[$frac(3, 2) #h(0.1em) z + frac(1, 3) = z − frac(2, 3)$]

#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")[$frac(1, 2) #h(0.1em) x − frac(1, 4) = frac(1, 12) #h(0.1em) x + frac(1, 6)$]

#solutionbox[
#emph[x] = 1
]

#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")[$frac(1, 2) #h(0.1em) a − frac(1, 4) = frac(1, 6) #h(0.1em) a + frac(1, 12)$]

#math.equation(block: true, alt: "the fraction 1 over 3 b plus the fraction 1 over 5 equals the fraction 2 over 5 b minus the fraction 3 over 5")[$frac(1, 3) #h(0.1em) b + frac(1, 5) = frac(2, 5) #h(0.1em) b − frac(3, 5)$]

#solutionbox[
#emph[b] = 12
]

#math.equation(block: true, alt: "the fraction 1 over 3 x plus the fraction 2 over 5 equals the fraction 1 over 5 x minus the fraction 2 over 5")[$frac(1, 3) #h(0.1em) x + frac(2, 5) = frac(1, 5) #h(0.1em) x − frac(2, 5)$]

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

#solutionbox[
#emph[x] = 1
]

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

#math.equation(block: true, alt: "the fraction 1 over 4 open parenthesis p minus 7 close parenthesis equals the fraction 1 over 3 open parenthesis p plus 5 close parenthesis")[$frac(1, 4) ( p − 7 ) = frac(1, 3) ( p + 5 )$]

#solutionbox[
#emph[p] = −41
]

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

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

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

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

#strong[Solve Equations with Decimal Coefficients]

In the following exercises, solve the equation by clearing the decimals.

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

#solutionbox[
#emph[y] = 10
]

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

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

#solutionbox[
#emph[j] = 2
]

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

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

#solutionbox[
#emph[x] = 18
]

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

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

#solutionbox[
#emph[x] = 18
]

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

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

#solutionbox[
#emph[x] = 20
]

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

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

#solutionbox[
#emph[n] = 9
]

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

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

#solutionbox[
#emph[d] = 8
]

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

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

#solutionbox[
#emph[q] = 11
]

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


==== Everyday Math

#strong[Coins] Taylor has #math.equation(block: false, alt: "$2.00")[$"$2.00"$] in dimes and pennies. The number of pennies is #math.equation(block: false, alt: "2")[$2$] 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.

#solutionbox[
#emph[d] = 18
]

#strong[Stamps] Travis bought #math.equation(block: false, alt: "$9.45")[$"$9.45"$] worth of #math.equation(block: false, alt: "49-cent")[$"49-cent"$] stamps and #math.equation(block: false, alt: "21-cent")[$"21-cent"$] stamps. The number of #math.equation(block: false, alt: "21-cent")[$"21-cent"$] stamps was #math.equation(block: false, alt: "5")[$5$] less than the number of #math.equation(block: false, alt: "49-cent")[$"49-cent"$] stamps. Solve the equation #math.equation(block: false, alt: "0.49 s plus 0.21 open parenthesis s minus 5 close parenthesis equals 9.45")[$0.49 s + 0.21 ( s − 5 ) = 9.45$] for #math.equation(block: false, alt: "s ,")[$s ,$] to find the number of #math.equation(block: false, alt: "49-cent")[$"49-cent"$] stamps Travis bought.


==== Writing Exercises

Explain how to find the least common denominator of #math.equation(block: false, alt: "the fraction 3 over 8 , the fraction 1 over 6 , and the fraction 2 over 3 .")[$frac(3, 8) , #h(0.2em) frac(1, 6) , #h(0.2em) "and" #h(0.2em) frac(2, 3) .$]

#solutionbox[
Answers will vary.
]

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

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

#solutionbox[
Answers will vary.
]

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?


==== Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

#figure(figph[A self-assessment chart for math skills, asking students to rate their ability to solve equations using a general strategy, with fraction coefficients, and with decimal coefficients, across three confidence levels.], alt: "A self-assessment chart for math skills, asking students to rate their ability to solve equations using a general strategy, with fraction coefficients, and with decimal coefficients, across three confidence levels.", caption: none)

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


=== Chapter Review Exercises


==== Solve Equations using the Subtraction and Addition Properties of Equality

In the following exercises, determine whether the given number is a solution to the equation.

#math.equation(block: true, alt: "x plus 16 equals 31 , x equals 15")[$x + 16 = 31 , #h(0.2em) x = 15$]

#solutionbox[
yes
]

#math.equation(block: true, alt: "w minus 8 equals 5 , w equals 3")[$w − 8 = 5 , #h(0.2em) w = 3$]

#math.equation(block: true, alt: "−9 n equals 45 , n equals 54")[$−9 n = 45 , #h(0.2em) n = 54$]

#solutionbox[
no
]

#math.equation(block: true, alt: "4 a equals 72 , a equals 18")[$4 a = 72 , #h(0.2em) a = 18$]

In the following exercises, solve the equation using the Subtraction Property of Equality.

#math.equation(block: true, alt: "x plus 7 equals 19")[$x + 7 = 19$]

#solutionbox[
12
]

#math.equation(block: true, alt: "y plus 2 equals −6")[$y + 2 = −6$]

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

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

#math.equation(block: true, alt: "n plus 3.6 equals 5.1")[$n + 3.6 = 5.1$]

In the following exercises, solve the equation using the Addition Property of Equality.

#math.equation(block: true, alt: "u minus 7 equals 10")[$u − 7 = 10$]

#solutionbox[
#emph[u] = 17
]

#math.equation(block: true, alt: "x minus 9 equals −4")[$x − 9 = −4$]

#math.equation(block: true, alt: "c minus the fraction 3 over 11 equals the fraction 9 over 11")[$c − frac(3, 11) = frac(9, 11)$]

#solutionbox[
#math.equation(block: true, alt: "c equals the fraction 12 over 11")[$c = frac(12, 11)$]
]

#math.equation(block: true, alt: "p minus 4.8 equals 14")[$p − 4.8 = 14$]

In the following exercises, solve the equation.

#math.equation(block: true, alt: "n minus 12 equals 32")[$n − 12 = 32$]

#solutionbox[
#emph[n] = 44
]

#math.equation(block: true, alt: "y plus 16 equals −9")[$y + 16 = −9$]

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

#solutionbox[
#math.equation(block: true, alt: "f equals the fraction 10 over 3")[$f = frac(10, 3)$]
]

#math.equation(block: true, alt: "d minus 3.9 equals 8.2")[$d − 3.9 = 8.2$]

#math.equation(block: true, alt: "y plus 8 minus 15 equals −3")[$y + 8 − 15 = −3$]

#solutionbox[
#emph[y] = 4
]

#math.equation(block: true, alt: "7 x plus 10 minus 6 x plus 3 equals 5")[$7 x + 10 − 6 x + 3 = 5$]

#math.equation(block: true, alt: "6 open parenthesis n minus 1 close parenthesis minus 5 n equals −14")[$6 ( n − 1 ) − 5 n = −14$]

#solutionbox[
#emph[n] = −8
]

#math.equation(block: true, alt: "8 open parenthesis 3 p plus 5 close parenthesis minus 23 open parenthesis p minus 1 close parenthesis equals 35")[$8 ( 3 p + 5 ) − 23 ( p − 1 ) = 35$]

In the following exercises, translate each English sentence into an algebraic equation and then solve it.

The sum of #math.equation(block: false, alt: "−6")[$−6$] and #math.equation(block: false, alt: "m")[$m$] is #math.equation(block: false, alt: "25 .")[$25 .$]

#solutionbox[
−6 + #emph[m] = 25; #emph[m] = 31
]

Four less than #math.equation(block: false, alt: "n")[$n$] is #math.equation(block: false, alt: "13 .")[$13 .$]

In the following exercises, translate into an algebraic equation and solve.

Rochelle’s daughter is #math.equation(block: false, alt: "11")[$11$] years old. Her son is #math.equation(block: false, alt: "3")[$3$] years younger. How old is her son?

#solutionbox[
#emph[s] = 11 − 3; 8 years old
]

Tan weighs #math.equation(block: false, alt: "146")[$146$] pounds. Minh weighs #math.equation(block: false, alt: "15")[$15$] pounds more than Tan. How much does Minh weigh?

Peter paid #math.equation(block: false, alt: "$9.75")[$"$9.75"$] to go to the movies, which was #math.equation(block: false, alt: "$46.25")[$"$46.25"$] less than he paid to go to a concert. How much did he pay for the concert?

#solutionbox[
#emph[c] − 46.25 = 9.75; \$56.00
]

Elissa earned #math.equation(block: false, alt: "$152.84")[$"$152.84"$] this week, which was #math.equation(block: false, alt: "$21.65")[$"$21.65"$] more than she earned last week. How much did she earn last week?


==== Solve Equations using the Division and Multiplication Properties of Equality

In the following exercises, solve each equation using the Division Property of Equality.

#math.equation(block: true, alt: "8 x equals 72")[$8 x = 72$]

#solutionbox[
#emph[x] = 9
]

#math.equation(block: true, alt: "13 a equals −65")[$13 a = −65$]

#math.equation(block: true, alt: "0.25 p equals 5.25")[$0.25 p = 5.25$]

#solutionbox[
#emph[p] = 21
]

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

In the following exercises, solve each equation using the Multiplication Property of Equality.

#math.equation(block: true, alt: "the fraction n over 6 equals 18")[$frac(n, 6) = 18$]

#solutionbox[
#emph[n] = 108
]

#math.equation(block: true, alt: "the fraction y over −10 equals 30")[$frac(y, −10) = 30$]

#math.equation(block: true, alt: "36 equals the fraction 3 over 4 x")[$36 = frac(3, 4) #h(0.1em) x$]

#solutionbox[
#emph[x] = 48
]

#math.equation(block: true, alt: "the fraction 5 over 8 u equals the fraction 15 over 16")[$frac(5, 8) #h(0.1em) u = frac(15, 16)$]

In the following exercises, solve each equation.

#math.equation(block: true, alt: "−18 m equals −72")[$−18 m = −72$]

#solutionbox[
#emph[m] = 4
]

#math.equation(block: true, alt: "the fraction c over 9 equals 36")[$frac(c, 9) = 36$]

#math.equation(block: true, alt: "0.45 x equals 6.75")[$0.45 x = 6.75$]

#solutionbox[
#emph[x] = 15
]

#math.equation(block: true, alt: "the fraction 11 over 12 equals the fraction 2 over 3 y")[$frac(11, 12) = frac(2, 3) #h(0.1em) y$]

#math.equation(block: true, alt: "5 r minus 3 r plus 9 r equals 35 minus 2")[$5 r − 3 r + 9 r = 35 − 2$]

#solutionbox[
#emph[r] = 3
]

#math.equation(block: true, alt: "24 x plus 8 x minus 11 x equals −7 −14")[$24 x + 8 x − 11 x = −7 −14$]


==== Solve Equations with Variables and Constants on Both Sides

In the following exercises, solve the equations with constants on both sides.

#math.equation(block: true, alt: "8 p plus 7 equals 47")[$8 p + 7 = 47$]

#solutionbox[
#emph[p] = 5
]

#math.equation(block: true, alt: "10 w minus 5 equals 65")[$10 w − 5 = 65$]

#math.equation(block: true, alt: "3 x plus 19 equals −47")[$3 x + 19 = −47$]

#solutionbox[
#emph[x] = −22
]

#math.equation(block: true, alt: "32 equals −4 minus 9 n")[$32 = −4 − 9 n$]

In the following exercises, solve the equations with variables on both sides.

#math.equation(block: true, alt: "7 y equals 6 y minus 13")[$7 y = 6 y − 13$]

#solutionbox[
#emph[y] = −13
]

#math.equation(block: true, alt: "5 a plus 21 equals 2 a")[$5 a + 21 = 2 a$]

#math.equation(block: true, alt: "k equals −6 k minus 35")[$k = −6 k − 35$]

#solutionbox[
#emph[k] = −5
]

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

In the following exercises, solve the equations with constants and variables on both sides.

#math.equation(block: true, alt: "12 x minus 9 equals 3 x plus 45")[$12 x − 9 = 3 x + 45$]

#solutionbox[
#emph[x] = 6
]

#math.equation(block: true, alt: "5 n minus 20 equals −7 n minus 80")[$5 n − 20 = −7 n − 80$]

#math.equation(block: true, alt: "4 u plus 16 equals −19 minus u")[$4 u + 16 = −19 − u$]

#solutionbox[
#emph[u] = −7
]

#math.equation(block: true, alt: "the fraction 5 over 8 c minus 4 equals the fraction 3 over 8 c plus 4")[$frac(5, 8) #h(0.1em) c − 4 = frac(3, 8) #h(0.1em) c + 4$]

In the following exercises, solve each linear equation using the general strategy.

#math.equation(block: true, alt: "6 open parenthesis x plus 6 close parenthesis equals 24")[$6 ( x + 6 ) = 24$]

#solutionbox[
#emph[x] = −2
]

#math.equation(block: true, alt: "9 open parenthesis 2 p minus 5 close parenthesis equals 72")[$9 ( 2 p − 5 ) = 72$]

#math.equation(block: true, alt: "minus open parenthesis s plus 4 close parenthesis equals 18")[$− ( s + 4 ) = 18$]

#solutionbox[
#emph[s] = −22
]

#math.equation(block: true, alt: "8 plus 3 open parenthesis n minus 9 close parenthesis equals 17")[$8 + 3 ( n − 9 ) = 17$]

#math.equation(block: true, alt: "23 minus 3 open parenthesis y minus 7 close parenthesis equals 8")[$23 − 3 ( y − 7 ) = 8$]

#solutionbox[
#emph[y] = 12
]

#math.equation(block: true, alt: "the fraction 1 over 3 open parenthesis 6 m plus 21 close parenthesis equals m minus 7")[$frac(1, 3) ( 6 m + 21 ) = m − 7$]

#math.equation(block: true, alt: "8 open parenthesis r minus 2 close parenthesis equals 6 open parenthesis r plus 10 close parenthesis")[$8 ( r − 2 ) = 6 ( r + 10 )$]

#solutionbox[
#emph[r] = 38
]

#math.equation(block: true, alt: "5 plus 7 open parenthesis 2 minus 5 x close parenthesis equals 2 open parenthesis 9 x plus 1 close parenthesis minus open parenthesis 13 x minus 57 close parenthesis")[$5 + 7 ( 2 − 5 x ) = 2 ( 9 x + 1 ) − ( 13 x − 57 )$]

#math.equation(block: true, alt: "4 open parenthesis 3.5 y plus 0.25 close parenthesis equals 365")[$4 ( 3.5 y + 0.25 ) = 365$]

#solutionbox[
#emph[y] = 26
]

#math.equation(block: true, alt: "0.25 open parenthesis q minus 8 close parenthesis equals 0.1 open parenthesis q plus 7 close parenthesis")[$0.25 ( q − 8 ) = 0.1 ( q + 7 )$]


==== Solve Equations with Fraction or Decimal Coefficients

In the following exercises, solve each equation by clearing the fractions.

#math.equation(block: true, alt: "the fraction 2 over 5 n minus the fraction 1 over 10 equals the fraction 7 over 10")[$frac(2, 5) #h(0.1em) n − frac(1, 10) = frac(7, 10)$]

#solutionbox[
#emph[n] = 2
]

#math.equation(block: true, alt: "the fraction 1 over 3 x plus the fraction 1 over 5 x equals 8")[$frac(1, 3) #h(0.1em) x + frac(1, 5) #h(0.1em) x = 8$]

#math.equation(block: true, alt: "the fraction 3 over 4 a minus the fraction 1 over 3 equals the fraction 1 over 2 a plus the fraction 5 over 6")[$frac(3, 4) #h(0.1em) a − frac(1, 3) = frac(1, 2) #h(0.1em) a + frac(5, 6)$]

#solutionbox[
#math.equation(block: true, alt: "a equals the fraction 14 over 3")[$a = frac(14, 3)$]
]

#math.equation(block: true, alt: "the fraction 1 over 2 open parenthesis k plus 3 close parenthesis equals the fraction 1 over 3 open parenthesis k plus 16 close parenthesis")[$frac(1, 2) ( k + 3 ) = frac(1, 3) ( k + 16 )$]

In the following exercises, solve each equation by clearing the decimals.

#math.equation(block: true, alt: "0.8 x minus 0.3 equals 0.7 x plus 0.2")[$0.8 x − 0.3 = 0.7 x + 0.2$]

#solutionbox[
#emph[x] = 5
]

#math.equation(block: true, alt: "0.36 u plus 2.55 equals 0.41 u plus 6.8")[$0.36 u + 2.55 = 0.41 u + 6.8$]

#math.equation(block: true, alt: "0.6 p minus 1.9 equals 0.78 p plus 1.7")[$0.6 p − 1.9 = 0.78 p + 1.7$]

#solutionbox[
#emph[p] = −20
]

#math.equation(block: true, alt: "0.10 d plus 0.05 open parenthesis d minus 4 close parenthesis equals 2.05")[$0.10 d + 0.05 ( d − 4 ) = 2.05$]


=== Chapter Practice Test

Determine whether each number is a solution to the equation. #linebreak() 
#math.equation(block: false, alt: "3 x plus 5 equals 23 .")[$3 x + 5 = 23 .$]

+ ⓐ #math.equation(block: false, alt: "6")[$#h(0.2em) 6$]
+ ⓑ #math.equation(block: false, alt: "the fraction 23 over 5")[$#h(0.2em) frac(23, 5)$]

#solutionbox[
+ ⓐ yes
+ ⓑ no
]

In the following exercises, solve each equation.

#math.equation(block: true, alt: "n minus 18 equals 31")[$n − 18 = 31$]

#math.equation(block: true, alt: "9 c equals 144")[$9 c = 144$]

#solutionbox[
#emph[c] = 16
]

#math.equation(block: true, alt: "4 y minus 8 equals 16")[$4 y − 8 = 16$]

#math.equation(block: true, alt: "−8 x minus 15 plus 9 x minus 1 equals −21")[$−8 x − 15 + 9 x − 1 = −21$]

#solutionbox[
#emph[x] = −5
]

#math.equation(block: true, alt: "−15 a equals 120")[$−15 a = 120$]

#math.equation(block: true, alt: "the fraction 2 over 3 x equals 6")[$frac(2, 3) #h(0.1em) x = 6$]

#solutionbox[
#emph[x] = 9
]

#math.equation(block: true, alt: "x plus 3.8 equals 8.2")[$x + 3.8 = 8.2$]

#math.equation(block: true, alt: "10 y equals −5 y plus 60")[$10 y = −5 y + 60$]

#solutionbox[
#emph[y] = 4
]

#math.equation(block: true, alt: "8 n plus 2 equals 6 n plus 12")[$8 n + 2 = 6 n + 12$]

#math.equation(block: true, alt: "9 m minus 2 minus 4 m plus m equals 42 minus 8")[$9 m − 2 − 4 m + m = 42 − 8$]

#solutionbox[
#emph[m] = 6
]

#math.equation(block: true, alt: "−5 open parenthesis 2 x plus 1 close parenthesis equals 45")[$−5 ( 2 x + 1 ) = 45$]

#math.equation(block: true, alt: "minus open parenthesis d plus 9 close parenthesis equals 23")[$− ( d + 9 ) = 23$]

#solutionbox[
#emph[d] = −32
]

#math.equation(block: true, alt: "the fraction 1 over 3 open parenthesis 6 m plus 21 close parenthesis equals m minus 7")[$frac(1, 3) ( 6 m + 21 ) = m − 7$]

#math.equation(block: true, alt: "2 open parenthesis 6 x plus 5 close parenthesis minus 8 equals −22")[$2 ( 6 x + 5 ) − 8 = −22$]

#solutionbox[
#emph[x] = −2
]

#math.equation(block: true, alt: "8 open parenthesis 3 a plus 5 close parenthesis minus 7 open parenthesis 4 a minus 3 close parenthesis equals 20 minus 3 a")[$8 ( 3 a + 5 ) − 7 ( 4 a − 3 ) = 20 − 3 a$]

#math.equation(block: true, alt: "the fraction 1 over 4 p plus the fraction 1 over 3 equals the fraction 1 over 2")[$frac(1, 4) #h(0.1em) p + frac(1, 3) = frac(1, 2)$]

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

#math.equation(block: true, alt: "0.1 d plus 0.25 open parenthesis d plus 8 close parenthesis equals 4.1")[$0.1 d + 0.25 ( d + 8 ) = 4.1$]

Translate and solve: The difference of twice #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "4")[$4$] is #math.equation(block: false, alt: "16 .")[$16 .$]

#solutionbox[
2#emph[x] − 4 = 16; #emph[x] = 10
]

Samuel paid #math.equation(block: false, alt: "$25.82")[$"$25.82"$] for gas this week, which was #math.equation(block: false, alt: "$3.47")[$"$3.47"$] less than he paid last week. How much did he pay last week?