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== 8.1#h(0.6em)Solve Equations Using the Subtraction and Addition Properties of Equality

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

Solve: #math.equation(block: false, alt: "n minus 12 equals 16 .")[$n − 12 = 16 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "28")[$28$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate into algebra ‘five less than #math.equation(block: false, alt: "x .’")[$x ".’"$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "x minus 5")[$x − 5$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Is #math.equation(block: false, alt: "x equals 2")[$x = 2$] a solution to #math.equation(block: false, alt: "5 x minus 3 equals 7 ?")[$5 x − 3 = 7 ?$] #linebreak() 
If you missed this problem, review .

#solutionbox[
yes
]
]

We are now ready to “get to the good stuff.” You have the basics down and are ready to begin one of the most important topics in algebra: solving equations. The applications are limitless and extend to all careers and fields. Also, the skills and techniques you learn here will help improve your critical thinking and problem-solving skills. This is a great benefit of studying mathematics and will be useful in your life in ways you may not see right now.


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

We began our work solving equations in previous chapters. It has been a while since we have seen an #strong[equation], so we will review some of the key concepts before we go any further.

We said that solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the #strong[variable] that make each side of the equation the same. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to the puzzle.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Solution of an Equation]

A #strong[solution of an equation] is a value of a variable that makes a true statement when substituted into the equation.
]

In the earlier sections, we listed the steps to determine if a value is a solution. We restate them here.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Determine whether a number is a solution to an equation.]

+ Substitute the number for the variable in the equation.
+ Simplify the expressions on both sides of the equation.
+ Determine whether the resulting equation is true.
- If it is true, the number is a solution.
- If it is not true, the number is not a solution.
]

#examplebox("Example 1")[][
Determine whether #math.equation(block: false, alt: "y equals the fraction 3 over 4")[$y = frac(3, 4)$] is a solution for #math.equation(block: false, alt: "4 y plus 3 equals 8 y .")[$4 y + 3 = 8 y .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[An image displaying the algebraic equation '4y + 3 = 8y' on a white background. The equation is rendered in a black, clear, serif-like font.], alt: "An image displaying the algebraic equation '4y + 3 = 8y' on a white background. The equation is rendered in a black, clear, serif-like font.", caption: none)]),
  [#figure(figph[The text 'Substitute 3/4 for y.' is displayed on a white background. The word 'Substitute' is in dark blue, the fraction '3/4' is in red, and 'for y.' is in dark blue.], alt: "The text 'Substitute 3/4 for y.' is displayed on a white background. The word 'Substitute' is in dark blue, the fraction '3/4' is in red, and 'for y.' is in dark blue.", caption: none)], [#figure(figph[A mathematical equation is displayed: 4(3/4) + 3 ?= 8(3/4). The question mark above the equals sign asks to verify if the statement is true. The numbers 3/4 are highlighted in red.], alt: "A mathematical equation is displayed: 4(3/4) + 3 ?= 8(3/4). The question mark above the equals sign asks to verify if the statement is true. The numbers 3/4 are highlighted in red.", caption: none)],
  [Multiply.], [#figure(figph[A mathematical equation '3 + 3 = 6' is displayed with a question mark above the equals sign, implying a query about its truth or an unknown.], alt: "A mathematical equation '3 + 3 = 6' is displayed with a question mark above the equals sign, implying a query about its truth or an unknown.", caption: none)],
  [Add.], [#figure(figph[The image shows the mathematical statement '6 = 6' followed by a checkmark, indicating that the equality is true. The text is rendered in a black sans-serif font against a plain white background.], alt: "The image shows the mathematical statement '6 = 6' followed by a checkmark, indicating that the equality is true. The text is rendered in a black sans-serif font against a plain white background.", caption: none)],
))

Since #math.equation(block: false, alt: "y equals the fraction 3 over 4")[$y = frac(3, 4)$] results in a true equation, #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$] is a solution to the equation #math.equation(block: false, alt: "4 y plus 3 equals 8 y .")[$4 y + 3 = 8 y .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Is #math.equation(block: false, alt: "y equals the fraction 2 over 3")[$y = frac(2, 3)$] a solution for #math.equation(block: false, alt: "9 y plus 2 equals 6 y ?")[$9 y + 2 = 6 y ?$]

#solutionbox[
no
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Is #math.equation(block: false, alt: "y equals the fraction 2 over 5")[$y = frac(2, 5)$] a solution for #math.equation(block: false, alt: "5 y minus 3 equals 10 y ?")[$5 y − 3 = 10 y ?$]

#solutionbox[
no
]
]

We introduced the Subtraction and Addition Properties of Equality in Solving Equations Using the Subtraction and
Addition Properties of Equality. In that section, we modeled how these properties work and then applied them to solving equations with whole numbers. We used these properties again each time we introduced a new system of numbers. Let’s review those properties here.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Subtraction and Addition Properties of Equality]

#strong[Subtraction Property of Equality]

For all real numbers #math.equation(block: false, alt: "a , b ,")[$a , b ,$] and #math.equation(block: false, alt: "c ,")[$c ,$] if #math.equation(block: false, alt: "a equals b ,")[$a = b ,$] then #math.equation(block: false, alt: "a minus c equals b minus c .")[$a − c = b − c .$]

#strong[Addition Property of Equality]

For all real numbers #math.equation(block: false, alt: "a , b ,")[$a , b ,$] and #math.equation(block: false, alt: "c ,")[$c ,$] if #math.equation(block: false, alt: "a equals b ,")[$a = b ,$] then #math.equation(block: false, alt: "a plus c equals b plus c .")[$a + c = b + c .$]
]

When you add or subtract the same quantity from both sides of an equation, you still have equality.

We introduced the #strong[Subtraction Property of Equality] earlier by modeling equations with envelopes and counters.  models the equation #math.equation(block: false, alt: "x plus 3 equals 8 .")[$x + 3 = 8 .$]

#figure(figph[An envelope and three yellow counters are shown on the left side. On the right side are eight yellow counters.], alt: "An envelope and three yellow counters are shown on the left side. On the right side are eight yellow counters.", caption: none)

The goal is to isolate the variable on one side of the equation. So we ‘took away’ #math.equation(block: false, alt: "3")[$3$] from both sides of the equation and found the #strong[solution] #math.equation(block: false, alt: "x equals 5 .")[$x = 5 .$]

Some people picture a balance scale, as in , when they solve equations.

#figure(figph[Three balance scales are shown. The top scale has one red weight on each side and is balanced. Beside it is “1 mass on each side equals balanced.” The next scale has two weights on each side and is balanced. Beside it is “2 masses on each side equals balanced.” The bottom scale has one weight on the left and two on the right. The right side is lower than the left. Beside the image is “1 mass on one side and 2 masses on the other equals unbalanced.”], alt: "Three balance scales are shown. The top scale has one red weight on each side and is balanced. Beside it is “1 mass on each side equals balanced.” The next scale has two weights on each side and is balanced. Beside it is “2 masses on each side equals balanced.” The bottom scale has one weight on the left and two on the right. The right side is lower than the left. Beside the image is “1 mass on one side and 2 masses on the other equals unbalanced.”", caption: none)

The quantities on both sides of the equal sign in an equation are equal, or balanced. Just as with the balance scale, whatever you do to one side of the equation you must also do to the other to keep it balanced.

Let’s review how to use Subtraction and Addition Properties of Equality to solve equations. We need to isolate the variable on one side of the equation. And we check our solutions by substituting the value into the equation to make sure we have a true statement.

#examplebox("Example 2")[][
Solve: #math.equation(block: false, alt: "x minus 11 equals −3 .")[$x − 11 = −3 .$]

#solutionbox[
To isolate #math.equation(block: false, alt: "x ,")[$x ,$] we undo the addition of #math.equation(block: false, alt: "11")[$11$] by using the Subtraction Property of Equality.

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[A mathematical equation is displayed, showing 'x - 11 = -3' in black characters against a white background.], alt: "A mathematical equation is displayed, showing 'x - 11 = -3' in black characters against a white background.", caption: none)], [],
  [We "undo" the subtraction of 11 by adding 11 to each side.], [#figure(figph[The equation x - 11 + 11 = -3 + 11, illustrating the step of adding 11 to both sides to solve for x. The added '11's are highlighted in red.], alt: "The equation x - 11 + 11 = -3 + 11, illustrating the step of adding 11 to both sides to solve for x. The added '11's are highlighted in red.", caption: none)], [],
  [Simplify.], [#figure(figph[The equation x = 8 is displayed in black text on a white background.], alt: "The equation x = 8 is displayed in black text on a white background.", caption: none)], [],
  [Check:], [#figure(figph[A mathematical equation is displayed, showing 'x - 11 = -3' in black text against a white background. This is a linear equation with one variable, 'x'.], alt: "A mathematical equation is displayed, showing 'x - 11 = -3' in black text against a white background. This is a linear equation with one variable, 'x'.", caption: none)], [],
  [Substitute #math.equation(block: false, alt: "x equals 8")[$x = 8$].], [#figure(figph[A mathematical equation reads 8 - 11 = -3, with a question mark positioned above the equals sign, suggesting a verification of its correctness.], alt: "A mathematical equation reads 8 - 11 = -3, with a question mark positioned above the equals sign, suggesting a verification of its correctness.", caption: none)], [],
  [], [#figure(figph[A mathematical equation shows '-3 = -3' with a black checkmark next to it on a white background, confirming the equality is correct.], alt: "A mathematical equation shows '-3 = -3' with a black checkmark next to it on a white background, confirming the equality is correct.", caption: none)], [],
))

Since #math.equation(block: false, alt: "x equals 8")[$x = 8$] makes #math.equation(block: false, alt: "x minus 11 equals −3")[$x − 11 = −3$] a true statement, we know that it is a solution to the equation.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "x plus 9 equals −7 .")[$x + 9 = −7 .$]

#solutionbox[
#emph[x] = −16
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "x plus 16 equals −4 .")[$x + 16 = −4 .$]

#solutionbox[
#emph[x] = −20
]
]

In the original equation in the previous example, #math.equation(block: false, alt: "11")[$11$] was added to the #math.equation(block: false, alt: "x")[$x$], so we subtracted #math.equation(block: false, alt: "11")[$11$] to ‘undo’ the addition. In the next example, we will need to ‘undo’ subtraction by using the #strong[Addition Property of Equality].

#examplebox("Example 3")[][
Solve: #math.equation(block: false, alt: "m plus 4 equals −5 .")[$m + 4 = −5 .$]

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[The equation m + 4 = -5 is displayed in black text on a white background.], alt: "The equation m + 4 = -5 is displayed in black text on a white background.", caption: none)], [],
  [Subtract 4 from each side to "undo" the addition.], [#figure(figph[A math problem demonstrating a step in solving for 'm', where 4 is subtracted from both sides of the equation m + 4 = -5. The subtractions are highlighted in red.], alt: "A math problem demonstrating a step in solving for 'm', where 4 is subtracted from both sides of the equation m + 4 = -5. The subtractions are highlighted in red.", caption: none)], [],
  [Simplify.], [#figure(figph[The image displays the algebraic equation m = -9, set against a plain white background.], alt: "The image displays the algebraic equation m = -9, set against a plain white background.", caption: none)], [],
  [Check:], [#figure(figph[The image displays the algebraic equation m + 4 = -5 in black text against a white background.], alt: "The image displays the algebraic equation m + 4 = -5 in black text against a white background.", caption: none)], [],
  [Substitute #math.equation(block: false, alt: "m equals −9")[$m = −9$].], [#figure(figph[A mathematical equation displays '-9 + 4' followed by an equals sign with a question mark above it, and then '-5', querying whether -9 plus 4 is equal to -5.], alt: "A mathematical equation displays '-9 + 4' followed by an equals sign with a question mark above it, and then '-5', querying whether -9 plus 4 is equal to -5.", caption: none)], [],
  [], [#figure(figph[A close-up shot of a mathematical equation '-5 = -5' followed by a checkmark, indicating correctness or verification.], alt: "A close-up shot of a mathematical equation '-5 = -5' followed by a checkmark, indicating correctness or verification.", caption: none)], [],
  [], [], [The solution to #math.equation(block: false, alt: "m minus 4 equals −5")[$m − 4 = −5$] is #math.equation(block: false, alt: "m equals −1")[$m = −1$].],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "n minus 6 equals −7 .")[$n − 6 = −7 .$]

#solutionbox[
#emph[n] = −1
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "x minus 5 equals −9 .")[$x − 5 = −9 .$]

#solutionbox[
#emph[x] = −4
]
]

Now let’s review solving equations with fractions.

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

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[A mathematical equation is displayed on a white background, which reads 'n - 3/8 = 1/2'. The variable 'n' is followed by a minus sign, then the fraction three-eighths, an equals sign, and finally the fraction one-half.], alt: "A mathematical equation is displayed on a white background, which reads 'n - 3/8 = 1/2'. The variable 'n' is followed by a minus sign, then the fraction three-eighths, an equals sign, and finally the fraction one-half.", caption: none)], [],
  [Use the Addition Property of Equality.], [#figure(figph[A mathematical equation shows 'n minus three-eighths plus three-eighths equals one-half plus three-eighths.' The plus signs and the last 'three-eighths' are in red, while the rest are black.], alt: "A mathematical equation shows 'n minus three-eighths plus three-eighths equals one-half plus three-eighths.' The plus signs and the last 'three-eighths' are in red, while the rest are black.", caption: none)], [],
  [Find the LCD to add the fractions on the right.], [#figure(figph[A mathematical equation showing 'n minus three-eighths plus three-eighths equals four-eighths plus three-eighths'.], alt: "A mathematical equation showing 'n minus three-eighths plus three-eighths equals four-eighths plus three-eighths'.", caption: none)], [],
  [Simplify], [#figure(figph[The mathematical equation n = 7/8 is displayed in the center of a white background.], alt: "The mathematical equation n = 7/8 is displayed in the center of a white background.", caption: none)], [],
  [Check:], [#figure(figph[A mathematical equation shows 'n minus three eighths equals one half'.], alt: "A mathematical equation shows 'n minus three eighths equals one half'.", caption: none)], [],
  [#figure(figph[The text reads 'Substitute n = 7/8.' The word 'Substitute' and the variable 'n' are in a dark teal color, while the fraction '7/8' is in red. The equality sign '=' is also in teal.], alt: "The text reads 'Substitute n = 7/8.' The word 'Substitute' and the variable 'n' are in a dark teal color, while the fraction '7/8' is in red. The equality sign '=' is also in teal.", caption: none)], [#figure(figph[A math equation reads '7/8 - 3/8 = ? 1/2'. The problem involves subtracting two fractions with the same denominator and comparing the result to one-half.], alt: "A math equation reads '7/8 - 3/8 = ? 1/2'. The problem involves subtracting two fractions with the same denominator and comparing the result to one-half.", caption: none)], [],
  [Subtract.], [#figure(figph[A mathematical equation showing the fractions 4/8 and 1/2 separated by a question mark, indicating a missing operator. The question asks what symbol should replace the question mark.], alt: "A mathematical equation showing the fractions 4/8 and 1/2 separated by a question mark, indicating a missing operator. The question asks what symbol should replace the question mark.", caption: none)], [],
  [Simplify.], [#figure(figph[The image displays the equation 1/2 = 1/2, followed by a checkmark, indicating that the equality is correct or verified.], alt: "The image displays the equation 1/2 = 1/2, followed by a checkmark, indicating that the equality is correct or verified.", caption: none)], [],
  [The solution checks.], [], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "p minus the fraction 1 over 3 equals the fraction 5 over 6 .")[$p − frac(1, 3) = frac(5, 6) .$]

#solutionbox[
#math.equation(block: true, alt: "p equals the fraction 7 over 6")[$p = frac(7, 6)$]
]
]

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

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

In Solve Equations with Decimals, we solved equations that contained decimals. We’ll review this next.

#examplebox("Example 5")[][
Solve #math.equation(block: false, alt: "a minus 3.7 equals 4.3 .")[$a − 3.7 = 4.3 .$]

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[A mathematical equation is displayed, reading 'a - 3.7 = 4.3'. The variable 'a' is followed by a minus sign, then the decimal number '3.7', an equals sign, and finally the decimal number '4.3'.], alt: "A mathematical equation is displayed, reading 'a - 3.7 = 4.3'. The variable 'a' is followed by a minus sign, then the decimal number '3.7', an equals sign, and finally the decimal number '4.3'.", caption: none)], [],
  [Use the Addition Property of Equality.], [#figure(figph[The equation a - 3.7 + 3.7 = 4.3 + 3.7 illustrates adding 3.7 (highlighted in red) to both sides to isolate the variable 'a', applying the addition property of equality.], alt: "The equation a - 3.7 + 3.7 = 4.3 + 3.7 illustrates adding 3.7 (highlighted in red) to both sides to isolate the variable 'a', applying the addition property of equality.", caption: none)], [],
  [Add.], [#figure(figph[The mathematical equation 'a = 8' is displayed on a white background, representing that the variable 'a' is equal to the number 8.], alt: "The mathematical equation 'a = 8' is displayed on a white background, representing that the variable 'a' is equal to the number 8.", caption: none)], [],
  [Check:], [#figure(figph[A mathematical equation shows 'a minus 3.7 equals 4.3' in black text on a white background.], alt: "A mathematical equation shows 'a minus 3.7 equals 4.3' in black text on a white background.", caption: none)], [],
  [Substitute #math.equation(block: false, alt: "a equals 8")[$a = 8$].], [#figure(figph[A math problem is displayed: 8 - 3.7 =? 4.3. The number '8' is colored red, and a question mark is placed directly above the equals sign, suggesting an inquiry into whether the subtraction equals 4.3.], alt: "A math problem is displayed: 8 - 3.7 =? 4.3. The number '8' is colored red, and a question mark is placed directly above the equals sign, suggesting an inquiry into whether the subtraction equals 4.3.", caption: none)], [],
  [Simplify.], [#figure(figph[The mathematical equation '4.3 = 4.3' is displayed in black text on a white background, accompanied by a black checkmark indicating correctness.], alt: "The mathematical equation '4.3 = 4.3' is displayed in black text on a white background, accompanied by a black checkmark indicating correctness.", caption: none)], [],
  [The solution checks.], [], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "b minus 2.8 equals 3.6 .")[$b − 2.8 = 3.6 .$]

#solutionbox[
#emph[b] = 6.4
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "c minus 6.9 equals 7.1 .")[$c − 6.9 = 7.1 .$]

#solutionbox[
#emph[c] = 14
]
]


=== Solve Equations That Need to Be Simplified

In the examples up to this point, we have been able to isolate the variable with just one operation. Many of the equations we encounter in algebra will take more steps to solve. Usually, we will need to simplify one or both sides of an equation before using the Subtraction or Addition Properties of Equality. You should always simplify as much as possible before trying to isolate the variable.

#examplebox("Example 6")[][
Solve: #math.equation(block: false, alt: "3 x minus 7 minus 2 x minus 4 equals 1 .")[$3 x − 7 − 2 x − 4 = 1 .$]

#solutionbox[
The left side of the equation has an expression that we should simplify before trying to isolate the variable. #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed against a white background, reading '3x - 7 - 2x - 4 = 1'.], alt: "A mathematical equation is displayed against a white background, reading '3x - 7 - 2x - 4 = 1'.", caption: none)]),
  [Rearrange the terms, using the Commutative Property of Addition.], [#figure(figph[A mathematical equation on a white background: 3x - 2x - 7 - 4 = 1.], alt: "A mathematical equation on a white background: 3x - 2x - 7 - 4 = 1.", caption: none)],
  [Combine like terms.], [#figure(figph[A mathematical equation is displayed with a white background. The equation reads 'x - 11 = 1' with the characters rendered in black.], alt: "A mathematical equation is displayed with a white background. The equation reads 'x - 11 = 1' with the characters rendered in black.", caption: none)],
  [Add 11 to both sides to isolate #math.equation(block: false, alt: "x")[$x$].], [#figure(figph[A mathematical equation x - 11 + 11 = 1 + 11 is presented. The terms + 11 on both sides of the equation are highlighted in red, while the rest of the numbers, variables, and operators are in black.], alt: "A mathematical equation x - 11 + 11 = 1 + 11 is presented. The terms + 11 on both sides of the equation are highlighted in red, while the rest of the numbers, variables, and operators are in black.", caption: none)],
  [Simplify.], [#figure(figph[The equation x = 12 is displayed in black text against a white background.], alt: "The equation x = 12 is displayed in black text against a white background.", caption: none)],
  [Check. #linebreak()  Substitute #math.equation(block: false, alt: "x equals 12")[$x = 12$] into the original equation. #linebreak() #figure(figph[The top line shows 3x minus 7 minus 2x minus 4 equals 1. Below this is 3 times a red 12 minus 7 minus 2 times a red 12 minus 4 equals 1. Next is 36 minus 7 minus 24 minus 4 equals 1. Below is 29 minus 24 minus 4 equals 1. Next is 5 minus 4 equals 1. Last is 1 equals 1.], alt: "The top line shows 3x minus 7 minus 2x minus 4 equals 1. Below this is 3 times a red 12 minus 7 minus 2 times a red 12 minus 4 equals 1. Next is 36 minus 7 minus 24 minus 4 equals 1. Below is 29 minus 24 minus 4 equals 1. Next is 5 minus 4 equals 1. Last is 1 equals 1.", caption: none)], [],
))

The solution checks.
]
]

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

#solutionbox[
#emph[y] = 15
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "6 z plus 5 minus 5 z minus 4 equals 3 .")[$6 z + 5 − 5 z − 4 = 3 .$]

#solutionbox[
#emph[z] = 2
]
]

#examplebox("Example 7")[][
Solve: #math.equation(block: false, alt: "3 open parenthesis n minus 4 close parenthesis minus 2 n equals −3 .")[$3 ( n − 4 ) − 2 n = −3 .$]

#solutionbox[
The left side of the equation has an expression that we should simplify.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed: 3(n - 4) - 2n = -3. It's an algebraic problem requiring solving for the variable 'n', potentially involving distribution and combining like terms.], alt: "A mathematical equation is displayed: 3(n - 4) - 2n = -3. It's an algebraic problem requiring solving for the variable 'n', potentially involving distribution and combining like terms.", caption: none)]),
  [Distribute on the left.], [#figure(figph[A mathematical equation is displayed with the expression 3n - 12 - 2n = -3, showing variables, constants, and operators in a standard algebraic format.], alt: "A mathematical equation is displayed with the expression 3n - 12 - 2n = -3, showing variables, constants, and operators in a standard algebraic format.", caption: none)],
  [Use the Commutative Property to rearrange terms.], [#figure(figph[A mathematical equation is displayed on a white background, which reads '3n - 2n - 12 = -3'.], alt: "A mathematical equation is displayed on a white background, which reads '3n - 2n - 12 = -3'.", caption: none)],
  [Combine like terms.], [#figure(figph[A simple algebraic equation is displayed, showing 'n - 12 = -3' with the variable 'n' to be solved for.], alt: "A simple algebraic equation is displayed, showing 'n - 12 = -3' with the variable 'n' to be solved for.", caption: none)],
  [Isolate #emph[n] using the Addition Property of Equality.], [#figure(figph[The mathematical equation n - 12 + 12 = -3 + 12, showing 12 added to both sides in red to solve for n.], alt: "The mathematical equation n - 12 + 12 = -3 + 12, showing 12 added to both sides in red to solve for n.", caption: none)],
  [Simplify.], [#figure(figph[The mathematical equation 'n=9' is displayed in bold black text on a white background. The letter 'n' is lowercase and italicized, followed by an equals sign and the numeral '9'.], alt: "The mathematical equation 'n=9' is displayed in bold black text on a white background. The letter 'n' is lowercase and italicized, followed by an equals sign and the numeral '9'.", caption: none)],
  [Check.  #linebreak() Substitute #math.equation(block: false, alt: "n equals 9")[$n = 9$] into the original equation. #linebreak() 
#figure(figph[The top line says 3 times parentheses n minus 4 minus 2n equals negative 3. The next line says 3 times parentheses red 9 minus 3 minus 2 times red 9 equals negative 3. The next line says 3 times 5 minus 18 equals negative 3. Below this is 15 minus 18 equals negative 3. Last is negative 3 equals negative 3.], alt: "The top line says 3 times parentheses n minus 4 minus 2n equals negative 3. The next line says 3 times parentheses red 9 minus 3 minus 2 times red 9 equals negative 3. The next line says 3 times 5 minus 18 equals negative 3. Below this is 15 minus 18 equals negative 3. Last is negative 3 equals negative 3.", caption: none)

 #linebreak()  The solution checks.], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "5 open parenthesis p minus 3 close parenthesis minus 4 p equals −10 .")[$5 ( p − 3 ) − 4 p = −10 .$]

#solutionbox[
#emph[p] = 5
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "4 open parenthesis q plus 2 close parenthesis minus 3 q equals −8 .")[$4 ( q + 2 ) − 3 q = −8 .$]

#solutionbox[
#emph[q] = −16
]
]

#examplebox("Example 8")[][
Solve: #math.equation(block: false, alt: "2 open parenthesis 3 k minus 1 close parenthesis minus 5 k equals −2 minus 7 .")[$2 ( 3 k − 1 ) − 5 k = −2 − 7 .$]

#solutionbox[
Both sides of the equation have expressions that we should simplify before we isolate the variable.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed: 2(3k - 1) - 5k = -2 - 7.], alt: "A mathematical equation is displayed: 2(3k - 1) - 5k = -2 - 7.", caption: none)]),
  [Distribute on the left, subtract on the right.], [#figure(figph[A mathematical equation is displayed, showing '6k - 2 - 5k = -9'. This is an algebraic expression involving the variable 'k' that needs to be solved.], alt: "A mathematical equation is displayed, showing '6k - 2 - 5k = -9'. This is an algebraic expression involving the variable 'k' that needs to be solved.", caption: none)],
  [Use the Commutative Property of Addition.], [#figure(figph[The image shows the algebraic equation 6k - 5k - 2 = -9, which simplifies to k - 2 = -9.], alt: "The image shows the algebraic equation 6k - 5k - 2 = -9, which simplifies to k - 2 = -9.", caption: none)],
  [Combine like terms.], [#figure(figph[An algebraic equation showing 'k - 2 = -9' against a white background.], alt: "An algebraic equation showing 'k - 2 = -9' against a white background.", caption: none)],
  [Undo subtraction by using the Addition Property of Equality.], [#figure(figph[An algebraic equation k - 2 + 2 = -9 + 2, illustrating the step of adding 2 to both sides to solve for k.], alt: "An algebraic equation k - 2 + 2 = -9 + 2, illustrating the step of adding 2 to both sides to solve for k.", caption: none)],
  [Simplify.], [#figure(figph[The equation k = -7 is displayed in black text on a white background.], alt: "The equation k = -7 is displayed in black text on a white background.", caption: none)],
  [Check. #linebreak() Let #math.equation(block: false, alt: "k equals −7 .")[$k = −7 .$] #linebreak() 
#figure(figph[The top line says 2 times parentheses 3k minus 1 minus 5k equals negative 2 minus 7. Below this is 2 times parentheses red negative 7 minus 1 minus 5 times red negative 7 equals negative 2 minus 7. The next line says 2 times parentheses negative 21 minus 1 minus 5 times negative 7 equals negative 9. Below that is 2 times negative 22 plus 35 equals negative 9. Next is negative 44 plus 35 equals negative 9. The last line says negative 9 equals negative 9.], alt: "The top line says 2 times parentheses 3k minus 1 minus 5k equals negative 2 minus 7. Below this is 2 times parentheses red negative 7 minus 1 minus 5 times red negative 7 equals negative 2 minus 7. The next line says 2 times parentheses negative 21 minus 1 minus 5 times negative 7 equals negative 9. Below that is 2 times negative 22 plus 35 equals negative 9. Next is negative 44 plus 35 equals negative 9. The last line says negative 9 equals negative 9.", caption: none)

 #linebreak() The solution checks.], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "4 open parenthesis 2 h minus 3 close parenthesis minus 7 h equals −6 minus 7 .")[$4 ( 2 h − 3 ) − 7 h = −6 − 7 .$]

#solutionbox[
#emph[h] = −1
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "2 open parenthesis 5 x plus 2 close parenthesis minus 9 x equals −2 plus 7 .")[$2 ( 5 x + 2 ) − 9 x = −2 + 7 .$]

#solutionbox[
#emph[x] = 1
]
]


=== Translate an Equation and Solve

In previous chapters, we translated word sentences into equations. The first step is to look for the word (or words) that translate(s) to the equal sign.  reminds us of some of the words that translate to the equal sign.

#figure(table(
  columns: 7, align: left, inset: 6pt,
  [Equals (=)], [], [], [], [], [], [],
  [is], [is equal to], [is the same as], [the result is], [gives], [was], [will be],
))

Let’s review the steps we used to translate a sentence into an equation.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Translate a word sentence to an algebraic equation.]

+ Locate the "equals" word(s). Translate to an equal sign.
+ Translate the words to the left of the "equals" word(s) into an algebraic expression.
+ Translate the words to the right of the "equals" word(s) into an algebraic expression.
]

Now we are ready to try an example.

#examplebox("Example 9")[][
Translate and solve: five more than #math.equation(block: false, alt: "x")[$x$] is equal to #math.equation(block: false, alt: "26 .")[$26 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Translate.], [#figure(figph[The image translates the phrase 'Five more than x is equal to 26' into the algebraic equation 'x + 5 = 26', using brackets to show the corresponding parts.], alt: "The image translates the phrase 'Five more than x is equal to 26' into the algebraic equation 'x + 5 = 26', using brackets to show the corresponding parts.", caption: none)]),
  [Subtract 5 from both sides.], [#figure(figph[A mathematical equation shows 'x + 5 - 5 = 26 - 5' with the subtracted '5' highlighted in red on both sides of the equality sign, illustrating a step in solving for x.], alt: "A mathematical equation shows 'x + 5 - 5 = 26 - 5' with the subtracted '5' highlighted in red on both sides of the equality sign, illustrating a step in solving for x.", caption: none)],
  [Simplify.], [#figure(figph[A simple mathematical equation 'x = 21' is displayed in black text against a plain white background, centered within the frame.], alt: "A simple mathematical equation 'x = 21' is displayed in black text against a plain white background, centered within the frame.", caption: none)],
  [Check: #linebreak() Is #math.equation(block: false, alt: "26")[$26$] five more than #math.equation(block: false, alt: "21")[$21$]? #linebreak() #figure(figph[A mathematical equation is displayed: '21 + 5 =? 26'. The question mark above the equals sign indicates that the equation is a query, asking if 21 plus 5 is equal to 26.], alt: "A mathematical equation is displayed: '21 + 5 =? 26'. The question mark above the equals sign indicates that the equation is a query, asking if 21 plus 5 is equal to 26.", caption: none)

 #linebreak() #figure(figph[The image displays the equation '26 = 26' followed by a checkmark, indicating that the statement is correct.], alt: "The image displays the equation '26 = 26' followed by a checkmark, indicating that the statement is correct.", caption: none)

 #linebreak() The solution checks.], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve: Eleven more than #math.equation(block: false, alt: "x")[$x$] is equal to #math.equation(block: false, alt: "41 .")[$41 .$]

#solutionbox[
#emph[x] + 11 = 41; #emph[x] = 30
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve: Twelve less than #math.equation(block: false, alt: "y")[$y$] is equal to #math.equation(block: false, alt: "51 .")[$51 .$]

#solutionbox[
#emph[y] − 12 = 51; #emph[y] = 63
]
]

#examplebox("Example 10")[][
Translate and solve: The difference of #math.equation(block: false, alt: "5 p")[$5 p$] and #math.equation(block: false, alt: "4 p")[$4 p$] is #math.equation(block: false, alt: "23 .")[$23 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Translate.], [#figure(figph[Translating the word problem 'The difference of 5p and 4p is 23' into the algebraic equation 5p - 4p = 23.], alt: "Translating the word problem 'The difference of 5p and 4p is 23' into the algebraic equation 5p - 4p = 23.", caption: none)]),
  [Simplify.], [#figure(figph[The image shows the mathematical equation 'p = 23' written in black font on a plain white background, isolated in the center-right of the frame.], alt: "The image shows the mathematical equation 'p = 23' written in black font on a plain white background, isolated in the center-right of the frame.", caption: none)],
  [Check: #linebreak() #figure(figph[The image shows a mathematical equation: 5p - 4p = 23. This equation simplifies to p = 23.], alt: "The image shows a mathematical equation: 5p - 4p = 23. This equation simplifies to p = 23.", caption: none)

 #linebreak() #figure(figph[A math problem showing the expression 5(23) - 4(23) =? 23, which simplifies to 23 = 23, demonstrating a basic algebraic property with numerical values.], alt: "A math problem showing the expression 5(23) - 4(23) =? 23, which simplifies to 23 = 23, demonstrating a basic algebraic property with numerical values.", caption: none)

 #linebreak() #figure(figph[A mathematical expression is displayed: 115 minus 92, followed by an equals sign with a question mark above it, and then the number 23. It prompts to verify if 115 - 92 is indeed equal to 23.], alt: "A mathematical expression is displayed: 115 minus 92, followed by an equals sign with a question mark above it, and then the number 23. It prompts to verify if 115 - 92 is indeed equal to 23.", caption: none)

 #linebreak() #figure(figph[A mathematical equation '23 = 23' is prominently displayed on a white background, followed by a clear checkmark, indicating its correctness.], alt: "A mathematical equation '23 = 23' is prominently displayed on a white background, followed by a clear checkmark, indicating its correctness.", caption: none)], [],
  [The solution checks.], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve: The difference of #math.equation(block: false, alt: "4 x")[$4 x$] and #math.equation(block: false, alt: "3 x")[$3 x$] is #math.equation(block: false, alt: "14 .")[$14 .$]

#solutionbox[
4#emph[x] − 3#emph[x] = 14; #emph[x] = 14
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve: The difference of #math.equation(block: false, alt: "7 a")[$7 a$] and #math.equation(block: false, alt: "6 a")[$6 a$] is #math.equation(block: false, alt: "−8 .")[$−8 .$]

#solutionbox[
7#emph[a] − 6#emph[a] = −8; #emph[a] = −8
]
]


=== Translate and Solve Applications

In most of the application problems we solved earlier, we were able to find the quantity we were looking for by simplifying an algebraic expression. Now we will be using equations to solve application problems. We’ll start by restating the problem in just one sentence, assign a variable, and then translate the sentence into an equation to solve. When assigning a variable, choose a letter that reminds you of what you are looking for.

#examplebox("Example 11")[][
The Robles family has two dogs, Buster and Chandler. Together, they weigh #math.equation(block: false, alt: "71")[$71$] pounds.

Chandler weighs #math.equation(block: false, alt: "28")[$28$] pounds. How much does Buster weigh?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Read the problem carefully.], []),
  [Identify what you are asked to find, and choose a variable to represent it.], [How much does Buster weigh? #linebreak() 
   Let #math.equation(block: false, alt: "b equals")[$b =$] Buster's weight],
  [Write a sentence that gives the information to find it.], [Buster's weight plus Chandler's weight equals 71 pounds.],
  [We will restate the problem, and then include the given information.], [Buster's weight plus 28 equals 71.],
  [Translate the sentence into an equation, using the variable #math.equation(block: false, alt: "b")[$b$].], [#figure(figph[The image displays the algebraic equation b + 28 = 71, presented in black text on a white background.], alt: "The image displays the algebraic equation b + 28 = 71, presented in black text on a white background.", caption: none)],
  [Solve the equation using good algebraic techniques.], [#figure(figph[A mathematical equation is displayed on a white background, which reads 'b + 28 - 28 = 71 - 28'.], alt: "A mathematical equation is displayed on a white background, which reads 'b + 28 - 28 = 71 - 28'.", caption: none)

 #linebreak() 
  #figure(figph[The image displays the equation 'b = 43' in black text against a white background.], alt: "The image displays the equation 'b = 43' in black text against a white background.", caption: none)],
  [Check the answer in the problem and make sure it makes sense.], [],
  [Is 43 pounds a reasonable weight for a dog? Yes. Does Buster's weight plus Chandler's weight equal 71 pounds?], [],
  [#math.equation(block: false, alt: "43 plus 28 equals ? 71")[$43 + 28 limits(=)^(?) 71$]], [],
  [#math.equation(block: false, alt: "71 equals 71 ✓")[$71 = 71 #h(0.2em) ✓$]], [],
  [Write a complete sentence that answers the question, "How much does Buster weigh?"], [Buster weighs 43 pounds],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate into an algebraic equation and solve: The Pappas family has two cats, Zeus and Athena. Together, they weigh #math.equation(block: false, alt: "13")[$13$] pounds. Zeus weighs #math.equation(block: false, alt: "6")[$6$] pounds. How much does Athena weigh?

#solutionbox[
#emph[a] + 6 = 13; Athena weighs 7 pounds.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate into an algebraic equation and solve: Sam and Henry are roommates. Together, they have #math.equation(block: false, alt: "68")[$68$] books. Sam has #math.equation(block: false, alt: "26")[$26$] books. How many books does Henry have?

#solutionbox[
26 + #emph[h] = 68; Henry has 42 books.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Devise a problem-solving strategy.]

+ Read the problem. Make sure you understand all the words and ideas.
+ Identify what you are looking for.
+ Name what you are looking for. Choose a variable to represent that quantity.
+ Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
+ Solve the equation using good algebra techniques.
+ Check the answer in the problem and make sure it makes sense.
+ Answer the question with a complete sentence.
]

#examplebox("Example 12")[][
Shayla paid #math.equation(block: false, alt: "$24,575")[$"$24,575"$] for her new car. This was #math.equation(block: false, alt: "$875")[$"$875"$] less than the sticker price. What was the sticker price of the car?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([What are you asked to find?], ["What was the sticker price of the car?"]),
  [Assign a variable.], [Let #math.equation(block: false, alt: "s equals")[$s =$] the sticker price of the car.],
  [Write a sentence that gives the information to find it.], [\$24,575 is \$875 less than the sticker price #linebreak() 
   \$24,575 is \$875 less than #math.equation(block: false, alt: "s")[$s$]],
  [Translate into an equation.], [#figure(figph[A mathematical equation is displayed with the numbers 24,575 and 875, separated by an equals sign and a subtraction operation involving the variable 's'. The equation reads as 24,575 = s - 875.], alt: "A mathematical equation is displayed with the numbers 24,575 and 875, separated by an equals sign and a subtraction operation involving the variable 's'. The equation reads as 24,575 = s - 875.", caption: none)],
  [Solve.], [#figure(figph[An algebraic equation: 24,575 + 875 = s - 875 + 875. The right side simplifies to 's', so the equation becomes 24,575 + 875 = s.], alt: "An algebraic equation: 24,575 + 875 = s - 875 + 875. The right side simplifies to 's', so the equation becomes 24,575 + 875 = s.", caption: none)

 #linebreak() 
  #figure(figph[The number 25,450 is shown followed by an equals sign and the letter 's', forming the equation 25,450 = s.], alt: "The number 25,450 is shown followed by an equals sign and the letter 's', forming the equation 25,450 = s.", caption: none)],
  [Check:], [],
  [Is \$875 less than \$25,450 equal to \$24,575?], [],
  [#math.equation(block: false, alt: "25,450 minus 875 equals ? 24,575")[$25,450 − 875 limits(=)^(?) 24,575$]], [],
  [#math.equation(block: false, alt: "24,575 equals 24,575 ✓")[$24,575 = 24,575 #h(0.2em) ✓$]], [],
  [Write a sentence that answers the question.], [The sticker price was \$25,450.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate into an algebraic equation and solve: Eddie paid #math.equation(block: false, alt: "$19,875")[$"$19,875"$] for his new car. This was #math.equation(block: false, alt: "$1,025")[$"$1,025"$] less than the sticker price. What was the sticker price of the car?

#solutionbox[
19,875 = #emph[s] − 1025; the sticker price is \$20,900.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate into an algebraic equation and solve: The admission price for the movies during the day is #math.equation(block: false, alt: "$7.75 .")[$"$7.75" .$] This is #math.equation(block: false, alt: "$3.25")[$"$3.25"$] less than the price at night. How much does the movie cost at night?

#solutionbox[
7.75 = #emph[n] − 3.25; the price at night is \$11.00.
]
]

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

- #link("https://www.openstax.org/l/24Solveonestep")[Solving One Step Equations By Addition and Subtraction]
- #link("https://www.openstax.org/l/24SolveByAdd")[Solve One Step Equations By Add and Subtract Whole Numbers (Variable on Left)]
- #link("https://www.openstax.org/l/24AddSubtrWhole")[Solve One Step Equations By Add and Subtract Whole Numbers (Variable on Right)]
]


=== Key Concepts

- #emph[Determine whether a number is a solution to an equation.]+ Substitute the number for the variable in the equation.
+ Simplify the expressions on both sides of the equation.
+ Determine whether the resulting equation is true.


If it is true, the number is a solution. #linebreak() 
If it is not true, the number is not a solution.
- #emph[Subtraction and Addition Properties of Equality]- #emph[Subtraction Property of Equality] #linebreak() For all real numbers #emph[a, b,] and #emph[c], #linebreak() if #emph[a = b] then #math.equation(block: false, alt: "a minus c equals b minus c")[$a − c = b − c$].
- #emph[Addition Property of Equality] #linebreak() For all real numbers #emph[a, b,] and #emph[c], #linebreak() if #emph[a = b] then #math.equation(block: false, alt: "a plus c equals b plus c")[$a + c = b + c$].
- #emph[Translate a word sentence to an algebraic equation.]+ Locate the “equals” word(s). Translate to an equal sign.
+ Translate the words to the left of the “equals” word(s) into an algebraic expression.
+ Translate the words to the right of the “equals” word(s) into an algebraic expression.
- #emph[Problem-solving strategy]+ Read the problem. Make sure you understand all the words and ideas.
+ Identify what you are looking for.
+ Name what you are looking for. Choose a variable to represent that quantity.
+ Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
+ Solve the equation using good algebra techniques.
+ Check the answer in the problem and make sure it makes sense.
+ Answer the question with a complete sentence.


==== Practice Makes Perfect

#strong[Solve Equations Using the Subtraction and Addition Properties of Equality]

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

Is #math.equation(block: false, alt: "y equals the fraction 1 over 3")[$y = frac(1, 3)$] a solution of #math.equation(block: false, alt: "4 y plus 2 equals 10 y ?")[$4 y + 2 = 10 y ?$]

#solutionbox[
yes
]

Is #math.equation(block: false, alt: "x equals the fraction 3 over 4")[$x = frac(3, 4)$] a solution of #math.equation(block: false, alt: "5 x plus 3 equals 9 x ?")[$5 x + 3 = 9 x ?$]

Is #math.equation(block: false, alt: "u equals − the fraction 1 over 2")[$u = − frac(1, 2)$] a solution of #math.equation(block: false, alt: "8 u minus 1 equals 6 u ?")[$8 u − 1 = 6 u ?$]

#solutionbox[
no
]

Is #math.equation(block: false, alt: "v equals − the fraction 1 over 3")[$v = − frac(1, 3)$] a solution of #math.equation(block: false, alt: "9 v minus 2 equals 3 v ?")[$9 v − 2 = 3 v ?$]

In the following exercises, solve each equation.

#math.equation(block: true, alt: "x plus 7 equals 12")[$x + 7 = 12$]

#solutionbox[
#emph[x] = 5
]

#math.equation(block: true, alt: "y plus 5 equals −6")[$y + 5 = −6$]

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

#solutionbox[
#math.equation(block: true, alt: "b equals the fraction 1 over 2")[$b = frac(1, 2)$]
]

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

#math.equation(block: true, alt: "p plus 2.4 equals −9.3")[$p + 2.4 = −9.3$]

#solutionbox[
#emph[p] = −11.7
]

#math.equation(block: true, alt: "m plus 7.9 equals 11.6")[$m + 7.9 = 11.6$]

#math.equation(block: true, alt: "a minus 3 equals 7")[$a − 3 = 7$]

#solutionbox[
#emph[a] = 10
]

#math.equation(block: true, alt: "m minus 8 equals −20")[$m − 8 = −20$]

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

#solutionbox[
#math.equation(block: true, alt: "x equals the fraction 7 over 3")[$x = frac(7, 3)$]
]

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

#math.equation(block: true, alt: "y minus 3.8 equals 10")[$y − 3.8 = 10$]

#solutionbox[
#emph[y] = 13.8
]

#math.equation(block: true, alt: "y minus 7.2 equals 5")[$y − 7.2 = 5$]

#math.equation(block: true, alt: "x minus 15 equals −42")[$x − 15 = −42$]

#solutionbox[
#emph[x] = −27
]

#math.equation(block: true, alt: "z plus 5.2 equals −8.5")[$z + 5.2 = −8.5$]

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

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

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

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

#solutionbox[
#math.equation(block: true, alt: "y equals the fraction 27 over 20")[$y = frac(27, 20)$]
]

#strong[Solve Equations that Need to be Simplified]

In the following exercises, solve each equation.

#math.equation(block: true, alt: "c plus 3 minus 10 equals 18")[$c + 3 − 10 = 18$]

#math.equation(block: true, alt: "m plus 6 minus 8 equals 15")[$m + 6 − 8 = 15$]

#solutionbox[
#emph[m] = 17
]

#math.equation(block: true, alt: "9 x plus 5 minus 8 x plus 14 equals 20")[$9 x + 5 − 8 x + 14 = 20$]

#math.equation(block: true, alt: "6 x plus 8 minus 5 x plus 16 equals 32")[$6 x + 8 − 5 x + 16 = 32$]

#solutionbox[
#emph[x] = 8
]

#math.equation(block: true, alt: "−6 x minus 11 plus 7 x minus 5 equals −16")[$−6 x − 11 + 7 x − 5 = −16$]

#math.equation(block: true, alt: "−8 n minus 17 plus 9 n minus 4 equals −41")[$−8 n − 17 + 9 n − 4 = −41$]

#solutionbox[
#emph[n] = −20
]

#math.equation(block: true, alt: "3 open parenthesis y minus 5 close parenthesis minus 2 y equals −7")[$3 ( y − 5 ) − 2 y = −7$]

#math.equation(block: true, alt: "4 open parenthesis y minus 2 close parenthesis minus 3 y equals −6")[$4 ( y − 2 ) − 3 y = −6$]

#solutionbox[
#emph[y] = 2
]

#math.equation(block: true, alt: "8 open parenthesis u plus 1.5 close parenthesis minus 7 u equals 4.9")[$8 ( u + 1.5 ) − 7 u = 4.9$]

#math.equation(block: true, alt: "5 open parenthesis w plus 2.2 close parenthesis minus 4 w equals 9.3")[$5 ( w + 2.2 ) − 4 w = 9.3$]

#solutionbox[
#emph[w] = −1.7
]

#math.equation(block: true, alt: "−5 open parenthesis y minus 2 close parenthesis plus 6 y equals −7 plus 4")[$−5 ( y − 2 ) + 6 y = −7 + 4$]

#math.equation(block: true, alt: "−8 open parenthesis x minus 1 close parenthesis plus 9 x equals −3 plus 9")[$−8 ( x − 1 ) + 9 x = −3 + 9$]

#solutionbox[
#emph[x] = −2
]

#math.equation(block: true, alt: "3 open parenthesis 5 n minus 1 close parenthesis minus 14 n plus 9 equals 1 minus 2")[$3 ( 5 n − 1 ) − 14 n + 9 = 1 − 2$]

#math.equation(block: true, alt: "2 open parenthesis 8 m plus 3 close parenthesis minus 15 m minus 4 equals 3 minus 5")[$2 ( 8 m + 3 ) − 15 m − 4 = 3 − 5$]

#solutionbox[
#emph[m] = −4
]

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

#math.equation(block: true, alt: "minus open parenthesis k plus 7 close parenthesis plus 2 k plus 8 equals 7")[$− ( k + 7 ) + 2 k + 8 = 7$]

#solutionbox[
#emph[k] = 6
]

#math.equation(block: true, alt: "6 a minus 5 open parenthesis a minus 2 close parenthesis plus 9 equals −11")[$6 a − 5 ( a − 2 ) + 9 = −11$]

#math.equation(block: true, alt: "8 c minus 7 open parenthesis c minus 3 close parenthesis plus 4 equals −16")[$8 c − 7 ( c − 3 ) + 4 = −16$]

#solutionbox[
#emph[c] = −41
]

#math.equation(block: true, alt: "8 open parenthesis 4 x plus 5 close parenthesis minus 5 open parenthesis 6 x close parenthesis minus x equals 53")[$8 ( 4 x + 5 ) − 5 ( 6 x ) − x = 53$]

#math.equation(block: true, alt: "6 open parenthesis 9 y minus 1 close parenthesis minus 10 open parenthesis 5 y close parenthesis minus 3 y equals 22")[$6 ( 9 y − 1 ) − 10 ( 5 y ) − 3 y = 22$]

#solutionbox[
#emph[y] = 28
]

#strong[Translate to an Equation and Solve]

In the following exercises, translate to an equation and then solve.

Five more than #math.equation(block: false, alt: "x")[$x$] is equal to #math.equation(block: false, alt: "21 .")[$21 .$]

The sum of #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "−5")[$−5$] is #math.equation(block: false, alt: "33 .")[$33 .$]

#solutionbox[
#emph[x] + (−5) = 33; #emph[x] = 38
]

Ten less than #math.equation(block: false, alt: "m")[$m$] is #math.equation(block: false, alt: "−14 .")[$−14 .$]

Three less than #math.equation(block: false, alt: "y")[$y$] is #math.equation(block: false, alt: "−19 .")[$−19 .$]

#solutionbox[
#emph[y] − 3 = −19; #emph[y] = −16
]

The sum of #math.equation(block: false, alt: "y")[$y$] and #math.equation(block: false, alt: "−3")[$−3$] is #math.equation(block: false, alt: "40 .")[$40 .$]

Eight more than #math.equation(block: false, alt: "p")[$p$] is equal to #math.equation(block: false, alt: "52 .")[$52 .$]

#solutionbox[
#emph[p] + 8 = 52; #emph[p] = 44
]

The difference of #math.equation(block: false, alt: "9 x")[$9 x$] and #math.equation(block: false, alt: "8 x")[$8 x$] is #math.equation(block: false, alt: "17 .")[$17 .$]

The difference of #math.equation(block: false, alt: "5 c")[$5 c$] and #math.equation(block: false, alt: "4 c")[$4 c$] is #math.equation(block: false, alt: "60 .")[$60 .$]

#solutionbox[
5#emph[c] − 4#emph[c] = 60; #emph[c] = 60
]

The difference of #math.equation(block: false, alt: "n")[$n$] and #math.equation(block: false, alt: "the fraction 1 over 6")[$frac(1, 6)$] is #math.equation(block: false, alt: "the fraction 1 over 2 .")[$frac(1, 2) .$]

The difference of #math.equation(block: false, alt: "f")[$f$] and #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] is #math.equation(block: false, alt: "the fraction 1 over 12 .")[$frac(1, 12) .$]

#solutionbox[
#math.equation(block: true, alt: "f minus the fraction 1 over 3 equals the fraction 1 over 12 ; f equals the fraction 5 over 12")[$f − frac(1, 3) = frac(1, 12) ; #h(0.2em) f = frac(5, 12)$]
]

The sum of #math.equation(block: false, alt: "−4 n")[$−4 n$] and #math.equation(block: false, alt: "5 n")[$5 n$] is #math.equation(block: false, alt: "−32 .")[$−32 .$]

The sum of #math.equation(block: false, alt: "−9 m")[$−9 m$] and #math.equation(block: false, alt: "10 m")[$10 m$] is #math.equation(block: false, alt: "−25 .")[$−25 .$]

#solutionbox[
−9#emph[m] + 10#emph[m] = −25; #emph[m] = −25
]

#strong[Translate and Solve Applications]

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

Pilar drove from home to school and then to her aunt’s house, a total of #math.equation(block: false, alt: "18")[$18$] miles. The distance from Pilar’s house to school is #math.equation(block: false, alt: "7")[$7$] miles. What is the distance from school to her aunt’s house?

Jeff read a total of #math.equation(block: false, alt: "54")[$54$] pages in his English and Psychology textbooks. He read #math.equation(block: false, alt: "41")[$41$] pages in his English textbook. How many pages did he read in his Psychology textbook?

#solutionbox[
Let #emph[p] equal the number of pages read in the Psychology book. 41 + #emph[p] = 54. Jeff read 13 pages in his Psychology book.
]

Pablo’s father is #math.equation(block: false, alt: "3")[$3$] years older than his mother. Pablo’s mother is #math.equation(block: false, alt: "42")[$42$] years old. How old is his father?

Eva’s daughter is #math.equation(block: false, alt: "5")[$5$] years younger than her son. Eva’s son is #math.equation(block: false, alt: "12")[$12$] years old. How old is her daughter?

#solutionbox[
Let #emph[d] equal the daughter’s age. d = 12 − 5. Eva’s daughter’s age is 7 years old.
]

Allie weighs #math.equation(block: false, alt: "8")[$8$] pounds less than her twin sister Lorrie. Allie weighs #math.equation(block: false, alt: "124")[$124$] pounds. How much does Lorrie weigh?

For a family birthday dinner, Celeste bought a turkey that weighed #math.equation(block: false, alt: "5")[$5$] pounds less than the one she bought for Thanksgiving. The birthday dinner turkey weighed #math.equation(block: false, alt: "16")[$16$] pounds. How much did the Thanksgiving turkey weigh?

#solutionbox[
21 pounds
]

The nurse reported that Tricia’s daughter had gained #math.equation(block: false, alt: "4.2")[$4.2$] pounds since her last checkup and now weighs #math.equation(block: false, alt: "31.6")[$31.6$] pounds. How much did Tricia’s daughter weigh at her last checkup?

Connor’s temperature was #math.equation(block: false, alt: "0.7")[$0.7$] degrees higher this morning than it had been last night. His temperature this morning was #math.equation(block: false, alt: "101.2")[$101.2$] degrees. What was his temperature last night?

#solutionbox[
100.5 degrees
]

Melissa’s math book cost #math.equation(block: false, alt: "$22.85")[$"$22.85"$] less than her art book cost. Her math book cost #math.equation(block: false, alt: "$93.75 .")[$"$93.75" .$] How much did her art book cost?

Ron’s paycheck this week was #math.equation(block: false, alt: "$17.43")[$"$17.43"$] less than his paycheck last week. His paycheck this week was #math.equation(block: false, alt: "$103.76 .")[$"$103.76" .$] How much was Ron’s paycheck last week?

#solutionbox[
\$121.19
]


==== Everyday Math

#strong[Baking] Kelsey needs #math.equation(block: false, alt: "the fraction 2 over 3")[$frac(2, 3)$] cup of sugar for the cookie recipe she wants to make. She only has #math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$] cup of sugar and will borrow the rest from her neighbor. Let #math.equation(block: false, alt: "s")[$s$] equal the amount of sugar she will borrow. Solve the equation #math.equation(block: false, alt: "the fraction 1 over 4 plus s equals the fraction 2 over 3")[$frac(1, 4) + s = frac(2, 3)$] to find the amount of sugar she should ask to borrow.

#strong[Construction] Miguel wants to drill a hole for a #math.equation(block: false, alt: "the fraction 5 over 8 -inch")[$frac(5, "8") #h(0.1em) "-inch"$] screw. The screw should be #math.equation(block: false, alt: "the fraction 1 over 12")[$frac(1, 12)$] inch larger than the hole. Let #math.equation(block: false, alt: "d")[$d$] equal the size of the hole he should drill. Solve the equation #math.equation(block: false, alt: "d plus the fraction 1 over 12 equals the fraction 5 over 8")[$d + frac(1, 12) = frac(5, 8)$] to see what size the hole should be.

#solutionbox[
#math.equation(block: true, alt: "d equals the fraction 13 over 24")[$d = frac(13, 24)$]
]


==== Writing Exercises

Is #math.equation(block: false, alt: "−18")[$−18$] a solution to the equation #math.equation(block: false, alt: "3 x equals 16 minus 5 x ?")[$3 x = 16 − 5 x ?$] How do you know?

Write a word sentence that translates the equation #math.equation(block: false, alt: "y minus 18 equals 41")[$y − 18 = 41$] and then make up an application that uses this equation in its solution.

#solutionbox[
Answers will vary.
]


==== 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 with 'I can...' statements for solving equations, including using properties of equality, simplifying, translating, and solving applications. Options are 'Confidently,' 'With some help,' and 'No-I don't get it!'], alt: "A self-assessment chart with 'I can...' statements for solving equations, including using properties of equality, simplifying, translating, and solving applications. Options are 'Confidently,' 'With some help,' and 'No-I don't get it!'", caption: none)

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.