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== 8.3#h(0.6em)Solve Equations with Variables and Constants on Both Sides

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

Simplify: #math.equation(block: false, alt: "4 y minus 9 plus 9 .")[$4 y − 9 + 9 .$] #linebreak() 
If you missed this problem, review .

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "y plus 12 equals 16 .")[$y + 12 = 16 .$] #linebreak() 
If you missed this problem, review .

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "−3 y equals 63 .")[$−3 y = 63 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "minus 21")[$− 21$]
]
]


=== Solve an Equation with Constants on Both Sides

You may have noticed that in all the equations we have solved so far, all the #strong[variable] terms were on only one side of the equation with the #strong[constants] on the other side. This does not happen all the time—so now we’ll see how to solve equations where the variable terms and/or constant terms are on both sides of the equation.

Our strategy will involve choosing one side of the equation to be the variable side, and the other side of the equation to be the constant side. Then, we will use the Subtraction and Addition Properties of Equality, step by step, to get all the variable terms together on one side of the equation and the constant terms together on the other side.

By doing this, we will transform the equation that started with variables and constants on both sides into the form #math.equation(block: false, alt: "a x equals b .")[$a x = b .$] We already know how to solve equations of this form by using the Division or Multiplication Properties of Equality.

#examplebox("Example 1")[][
Solve: #math.equation(block: false, alt: "4 x plus 6 equals −14 .")[$4 x + 6 = −14 .$]

#solutionbox[
In this equation, the variable is only on the left side. It makes sense to call the left side the variable side. Therefore, the right side will be the constant side. We’ll write the labels above the equation to help us remember what goes where.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  [], [#figure(figph[The equation 4x + 6 = -14 is displayed, with 'variable' labeling 4x and 'constant' labeling 6 and -14, illustrating basic algebraic components.], alt: "The equation 4x + 6 = -14 is displayed, with 'variable' labeling 4x and 'constant' labeling 6 and -14, illustrating basic algebraic components.", caption: none)],
  [Since the left side is the variable side, the 6 is out of place. #linebreak()  We must "undo" adding 6 by subtracting 6, #linebreak()  and to keep the equality we must subtract 6 from both sides. #linebreak()  Use the Subtraction Property of Equality.], [#figure(figph[The equation 4x + 6 - 6 = -14 - 6, illustrating a step in solving for x by subtracting 6 from both sides, with the subtracted '6's highlighted in red.], alt: "The equation 4x + 6 - 6 = -14 - 6, illustrating a step in solving for x by subtracting 6 from both sides, with the subtracted '6's highlighted in red.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation is displayed on a white background: 4x = -20.], alt: "A mathematical equation is displayed on a white background: 4x = -20.", caption: none)],
  [Now all the #math.equation(block: false, alt: "x")[$x$]s are on the left and the constant on the right.], [],
  [Use the Division Property of Equality.], [#figure(figph[An algebraic equation showing 4x/4 = -20/4, with the denominator '4' in red, indicating division by 4 on both sides to solve for x.], alt: "An algebraic equation showing 4x/4 = -20/4, with the denominator '4' in red, indicating division by 4 on both sides to solve for x.", caption: none)],
  [Simplify.], [#figure(figph[The image displays the mathematical equation 'x = -5' in black font against a plain white background, centered in the frame. The 'x' is a lowercase variable, followed by an equals sign, and then a negative sign preceding the numeral '5'.], alt: "The image displays the mathematical equation 'x = -5' in black font against a plain white background, centered in the frame. The 'x' is a lowercase variable, followed by an equals sign, and then a negative sign preceding the numeral '5'.", caption: none)],
  [Check:], [#figure(figph[A mathematical equation is displayed: 4x + 6 = -14. The equation involves a variable 'x', addition, and negative numbers, representing a basic algebraic problem.], alt: "A mathematical equation is displayed: 4x + 6 = -14. The equation involves a variable 'x', addition, and negative numbers, representing a basic algebraic problem.", caption: none)],
  [Let #math.equation(block: false, alt: "x equals −5")[$x = −5$].], [#figure(figph[The image displays the mathematical equation 4(-5) + 6 = -14, showing the multiplication of 4 by negative 5, followed by the addition of 6, resulting in negative 14. The negative 5 is highlighted in red.], alt: "The image displays the mathematical equation 4(-5) + 6 = -14, showing the multiplication of 4 by negative 5, followed by the addition of 6, resulting in negative 14. The negative 5 is highlighted in red.", caption: none)],
  [], [#figure(figph[A mathematical equation is displayed on a white background, which reads '-20 + 6 = -14'. The numbers and symbols are in a dark gray font, creating a clear contrast.], alt: "A mathematical equation is displayed on a white background, which reads '-20 + 6 = -14'. The numbers and symbols are in a dark gray font, creating a clear contrast.", caption: none)],
  [], [#figure(figph[The equation -14 = -14 is correctly displayed with a checkmark.], alt: "The equation -14 = -14 is correctly displayed with a checkmark.", caption: none)],
))
]
]

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

#solutionbox[
#emph[x] = −4
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "5 a plus 3 equals −37 .")[$5 a + 3 = −37 .$]

#solutionbox[
#emph[a] = −8
]
]

#examplebox("Example 2")[][
Solve: #math.equation(block: false, alt: "2 y minus 7 equals 15 .")[$2 y − 7 = 15 .$]

#solutionbox[
Notice that the variable is only on the left side of the equation, so this will be the variable side and the right side will be the constant side. Since the left side is the variable side, the #math.equation(block: false, alt: "7")[$7$] is out of place. It is subtracted from the #math.equation(block: false, alt: "2 y ,")[$2 y ,$] so to ‘undo’ subtraction, add #math.equation(block: false, alt: "7")[$7$] to both sides.

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[An algebraic equation 2y - 7 = 15 with terms labeled as 'variable' (2y) and 'constant' (-7 and 15) to illustrate basic algebraic components.], alt: "An algebraic equation 2y - 7 = 15 with terms labeled as 'variable' (2y) and 'constant' (-7 and 15) to illustrate basic algebraic components.", caption: none)], [],
  [Add 7 to both sides.], [#figure(figph[The equation 2y - 7 + 7 = 15 + 7 illustrates the step of adding 7 to both sides to simplify the expression and solve for y, with the added 7 highlighted in red.], alt: "The equation 2y - 7 + 7 = 15 + 7 illustrates the step of adding 7 to both sides to simplify the expression and solve for y, with the added 7 highlighted in red.", caption: none)], [],
  [Simplify.], [#figure(figph[The mathematical equation '2y = 22' is displayed in black text on a white background.], alt: "The mathematical equation '2y = 22' is displayed in black text on a white background.", caption: none)], [],
  [The variables are now on one side and the constants on the other.], [], [],
  [Divide both sides by 2.], [#figure(figph[A mathematical equation showing 2y/2 = 22/2, with the denominator '2' highlighted in red on both sides.], alt: "A mathematical equation showing 2y/2 = 22/2, with the denominator '2' highlighted in red on both sides.", caption: none)], [],
  [Simplify.], [#figure(figph[The image displays a simple mathematical equation, 'y = 11', written in black text on a plain white background. The equation indicates that the variable 'y' is equal to the numerical value of 11.], alt: "The image displays a simple mathematical equation, 'y = 11', written in black text on a plain white background. The equation indicates that the variable 'y' is equal to the numerical value of 11.", caption: none)], [],
  [Check:], [#figure(figph[A mathematical equation, 2y - 7 = 15, is displayed in black text against a white background.], alt: "A mathematical equation, 2y - 7 = 15, is displayed in black text against a white background.", caption: none)], [],
  [Substitute: #math.equation(block: false, alt: "y equals 11")[$y = 11$].], [#figure(figph[A math problem asking if 2 \* 11 - 7 equals 15. The number 11 is highlighted in red. The equation is true: 22 - 7 = 15.], alt: "A math problem asking if 2 * 11 - 7 equals 15. The number 11 is highlighted in red. The equation is true: 22 - 7 = 15.", caption: none)], [],
  [], [#figure(figph[The mathematical expression '22 - 7 ?= 15' is shown, verifying if 22 minus 7 is equal to 15.], alt: "The mathematical expression '22 - 7 ?= 15' is shown, verifying if 22 minus 7 is equal to 15.", caption: none)], [],
  [], [#figure(figph[The equation '15 = 15' is displayed, followed by a checkmark, confirming its correctness.], alt: "The equation '15 = 15' is displayed, followed by a checkmark, confirming its correctness.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "5 y minus 9 equals 16 .")[$5 y − 9 = 16 .$]

#solutionbox[
#emph[y] = 5
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "3 m minus 8 equals 19 .")[$3 m − 8 = 19 .$]

#solutionbox[
#emph[m] = 9
]
]


=== Solve an Equation with Variables on Both Sides

What if there are variables on both sides of the equation? We will start like we did above—choosing a #strong[variable] side and a constant side, and then use the Subtraction and Addition Properties of Equality to collect all variables on one side and all #strong[constants] on the other side. Remember, what you do to the left side of the equation, you must do to the right side too.

#examplebox("Example 3")[][
Solve: #math.equation(block: false, alt: "5 x equals 4 x plus 7 .")[$5 x = 4 x + 7 .$]

#solutionbox[
Here the variable, #math.equation(block: false, alt: "x ,")[$x ,$] is on both sides, but the constants appear only on the right side, so let’s make the right side the “constant” side. Then the left side will be the “variable” side.

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[An image showing the equation 5x = 4x + 7, with 'variable' labeled above 5x and 'constant' labeled above 7.], alt: "An image showing the equation 5x = 4x + 7, with 'variable' labeled above 5x and 'constant' labeled above 7.", caption: none)], [],
  [We don't want any variables on the right, so subtract the #math.equation(block: false, alt: "4 x")[$4 x$].], [#figure(figph[An algebraic equation showing 5x minus 4x equals 4x minus 4x plus 7. The 4x terms are highlighted in red, indicating they are being subtracted or canceled out from both sides of the equation.], alt: "An algebraic equation showing 5x minus 4x equals 4x minus 4x plus 7. The 4x terms are highlighted in red, indicating they are being subtracted or canceled out from both sides of the equation.", caption: none)], [],
  [Simplify.], [#figure(figph[The mathematical equation x=7 is displayed in black font against a white background.], alt: "The mathematical equation x=7 is displayed in black font against a white background.", caption: none)], [],
  [We have all the variables on one side and the constants on the other. We have solved the equation.], [], [],
  [Check:], [#figure(figph[A basic algebraic equation is displayed, reading 5x = 4x + 7, shown in a bold, sans-serif font against a white background.], alt: "A basic algebraic equation is displayed, reading 5x = 4x + 7, shown in a bold, sans-serif font against a white background.", caption: none)], [],
  [Substitute 7 for #math.equation(block: false, alt: "x")[$x$].], [#figure(figph[A mathematical equation is displayed: 5(7) with a question mark over an equals sign, then 4(7) + 7. The number 7 is highlighted in red in each instance within parentheses.], alt: "A mathematical equation is displayed: 5(7) with a question mark over an equals sign, then 4(7) + 7. The number 7 is highlighted in red in each instance within parentheses.", caption: none)], [],
  [], [#figure(figph[A mathematical equation reads '35 =? 28 + 7'. The question mark above the equals sign indicates that the equation is being posed as a question or problem to be solved, asking if 35 is indeed equal to 28 plus 7.], alt: "A mathematical equation reads '35 =? 28 + 7'. The question mark above the equals sign indicates that the equation is being posed as a question or problem to be solved, asking if 35 is indeed equal to 28 plus 7.", caption: none)], [],
  [], [#figure(figph[A mathematical equation shows '35 = 35' with a checkmark, signifying that the statement is correct.], alt: "A mathematical equation shows '35 = 35' with a checkmark, signifying that the statement is correct.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "6 n equals 5 n plus 10 .")[$6 n = 5 n + 10 .$]

#solutionbox[
#emph[n] = 10
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "−6 c equals −7 c plus 1 .")[$−6 c = −7 c + 1 .$]

#solutionbox[
#emph[c] = 1
]
]

#examplebox("Example 4")[][
Solve: #math.equation(block: false, alt: "5 y minus 8 equals 7 y .")[$5 y − 8 = 7 y .$]

#solutionbox[
The only constant, #math.equation(block: false, alt: "−8 ,")[$−8 ,$] is on the left side of the equation and variable, #math.equation(block: false, alt: "y ,")[$y ,$] is on both sides. Let’s leave the constant on the left and collect the variables to the right.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The equation 5y - 8 = 7y is shown, with the word 'constant' in red text above '5y - 8', and 'variable' in red text above '7y'.], alt: "The equation 5y - 8 = 7y is shown, with the word 'constant' in red text above '5y - 8', and 'variable' in red text above '7y'.", caption: none)]),
  [Subtract #math.equation(block: false, alt: "5 y")[$5 y$] from both sides.], [#figure(figph[A mathematical equation is displayed: 5y - 5y - 8 = 7y - 5y. The terms -5y on the left side and -5y on the right side are highlighted in red, indicating they are part of a simplification step.], alt: "A mathematical equation is displayed: 5y - 5y - 8 = 7y - 5y. The terms -5y on the left side and -5y on the right side are highlighted in red, indicating they are part of a simplification step.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation is displayed with the expression -8 = 2y, representing a simple linear equation where the variable 'y' needs to be solved.], alt: "A mathematical equation is displayed with the expression -8 = 2y, representing a simple linear equation where the variable 'y' needs to be solved.", caption: none)],
  [We have the variables on the right and the constants on the left. Divide both sides by 2.], [#figure(figph[A mathematical equation shows '-8' divided by '2' (in red) equals '2y' divided by '2' (in red), indicating a step in solving for 'y' where both sides of the equation are divided by 2.], alt: "A mathematical equation shows '-8' divided by '2' (in red) equals '2y' divided by '2' (in red), indicating a step in solving for 'y' where both sides of the equation are divided by 2.", caption: none)],
  [Simplify.], [#figure(figph[The image displays the simple algebraic equation '-4 = y' against a white background.], alt: "The image displays the simple algebraic equation '-4 = y' against a white background.", caption: none)],
  [Rewrite with the variable on the left.], [#figure(figph[The equation y = -4 is displayed on a white background.], alt: "The equation y = -4 is displayed on a white background.", caption: none)],
  [Check: Let #math.equation(block: false, alt: "y equals −4")[$y = −4$].], [],
  [#figure(figph[A mathematical equation is displayed with black text on a white background, reading '5y - 8 = 7y'.], alt: "A mathematical equation is displayed with black text on a white background, reading '5y - 8 = 7y'.", caption: none)], [],
  [#figure(figph[A math equation reads 5(-4) - 8 =? 7(-4). Negative numbers are highlighted in red, suggesting a comparison or calculation of both sides.], alt: "A math equation reads 5(-4) - 8 =? 7(-4). Negative numbers are highlighted in red, suggesting a comparison or calculation of both sides.", caption: none)], [],
  [#figure(figph[A mathematical equation reads -20 - 8 = -28, with a question mark above the equals sign, asking if the statement is true.], alt: "A mathematical equation reads -20 - 8 = -28, with a question mark above the equals sign, asking if the statement is true.", caption: none)], [],
  [#figure(figph[A mathematical equation shows '-28 = -28' with a checkmark, indicating the equality is correct.], alt: "A mathematical equation shows '-28 = -28' with a checkmark, indicating the equality is correct.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "3 p minus 14 equals 5 p .")[$3 p − 14 = 5 p .$]

#solutionbox[
#emph[p] = −7
]
]

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

#solutionbox[
#emph[m] = −3
]
]

#examplebox("Example 5")[][
Solve: #math.equation(block: false, alt: "7 x equals minus x plus 24 .")[$7 x = − x + 24 .$]

#solutionbox[
The only constant, #math.equation(block: false, alt: "24 ,")[$24 ,$] is on the right, so let the left side be the variable side.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[An algebraic equation 7x = -x + 24 is displayed. The left side (7x) is labeled 'variable side' and the right side (-x + 24) is labeled 'constant side' in red text.], alt: "An algebraic equation 7x = -x + 24 is displayed. The left side (7x) is labeled 'variable side' and the right side (-x + 24) is labeled 'constant side' in red text.", caption: none)]),
  [Remove the #math.equation(block: false, alt: "minus x")[$− x$] from the right side by adding #math.equation(block: false, alt: "x")[$x$] to both sides.], [#figure(figph[A mathematical equation shows 7x + x = -x + x + 24. The plus and x symbols are highlighted in red on both sides of the equation.], alt: "A mathematical equation shows 7x + x = -x + x + 24. The plus and x symbols are highlighted in red on both sides of the equation.", caption: none)],
  [Simplify.], [#figure(figph[The mathematical equation '8x = 24' is displayed in bold black text on a white background.], alt: "The mathematical equation '8x = 24' is displayed in bold black text on a white background.", caption: none)],
  [All the variables are on the left and the constants are on the right. Divide both sides by 8.], [#figure(figph[The equation 8x/8 = 24/8, demonstrating dividing both sides by 8 (highlighted in red) to solve for x.], alt: "The equation 8x/8 = 24/8, demonstrating dividing both sides by 8 (highlighted in red) to solve for x.", caption: none)],
  [Simplify.], [#figure(figph[The mathematical equation 'x = 3' is displayed in black font against a plain white background.], alt: "The mathematical equation 'x = 3' is displayed in black font against a plain white background.", caption: none)],
  [Check: Substitute #math.equation(block: false, alt: "x equals 3")[$x = 3$].], [],
  [#figure(figph[The image demonstrates the verification of the solution to the algebraic equation 7x = -x + 24. By substituting x=3, the equation simplifies to 21 = 21, confirming that 3 is indeed the correct solution.], alt: "The image demonstrates the verification of the solution to the algebraic equation 7x = -x + 24. By substituting x=3, the equation simplifies to 21 = 21, confirming that 3 is indeed the correct solution.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "12 j equals −4 j plus 32 .")[$12 j = −4 j + 32 .$]

#solutionbox[
#emph[j] = 2
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "8 h equals −4 h plus 12 .")[$8 h = −4 h + 12 .$]

#solutionbox[
#emph[h] = 1
]
]


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

The next example will be the first to have variables #emph[and] constants on both sides of the equation. As we did before, we’ll collect the #strong[variable] terms to one side and the #strong[constants] to the other side.

#examplebox("Example 6")[][
Solve: #math.equation(block: false, alt: "7 x plus 5 equals 6 x plus 2 .")[$7 x + 5 = 6 x + 2 .$]

#solutionbox[
Start by choosing which side will be the variable side and which side will be the constant side. The variable terms are #math.equation(block: false, alt: "7 x")[$7 x$] and #math.equation(block: false, alt: "6 x .")[$6 x .$] Since #math.equation(block: false, alt: "7")[$7$] is greater than #math.equation(block: false, alt: "6 ,")[$6 ,$] make the left side the variable side and so the right side will be the constant side.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed, showing '7x + 5 = 6x + 2' in black text on a white background, representing a linear equation in one variable.], alt: "A mathematical equation is displayed, showing '7x + 5 = 6x + 2' in black text on a white background, representing a linear equation in one variable.", caption: none)]),
  [Collect the variable terms to the left side by subtracting #math.equation(block: false, alt: "6 x")[$6 x$] from both sides.], [#figure(figph[A mathematical equation, 7x - 6x + 5 = 6x - 6x + 2, is displayed. The terms '-6x' on the left side and '6x - 6x' on the right side are highlighted in red.], alt: "A mathematical equation, 7x - 6x + 5 = 6x - 6x + 2, is displayed. The terms '-6x' on the left side and '6x - 6x' on the right side are highlighted in red.", caption: none)],
  [Simplify.], [#figure(figph[The image shows a mathematical equation: x + 5 = 2.], alt: "The image shows a mathematical equation: x + 5 = 2.", caption: none)],
  [Now, collect the constants to the right side by subtracting 5 from both sides.], [#figure(figph[The equation x + 5 - 5 = 2 - 5 is shown, illustrating a step in solving for x by subtracting 5 from both sides, with the subtracted 5s in red.], alt: "The equation x + 5 - 5 = 2 - 5 is shown, illustrating a step in solving for x by subtracting 5 from both sides, with the subtracted 5s in red.", caption: none)],
  [Simplify.], [#figure(figph[The equation x = -3 is displayed in black font against a white background.], alt: "The equation x = -3 is displayed in black font against a white background.", caption: none)],
  [The solution is #math.equation(block: false, alt: "x equals −3")[$x = −3$].], [],
  [Check: Let #math.equation(block: false, alt: "x equals −3")[$x = −3$].], [],
  [#figure(figph[Step-by-step solution and verification of the linear equation 7x + 5 = 6x + 2, demonstrating that x = -3 is the correct solution by substituting it back into the equation.], alt: "Step-by-step solution and verification of the linear equation 7x + 5 = 6x + 2, demonstrating that x = -3 is the correct solution by substituting it back into the equation.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "12 x plus 8 equals 6 x plus 2 .")[$12 x + 8 = 6 x + 2 .$]

#solutionbox[
#emph[x] = −1
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "9 y plus 4 equals 7 y plus 12 .")[$9 y + 4 = 7 y + 12 .$]

#solutionbox[
#emph[y] = 4
]
]

We’ll summarize the steps we took so you can easily refer to them.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Solve an equation with variables and constants on both sides.]

+ Choose one side to be the variable side and then the other will be the constant side.
+ Collect the variable terms to the variable side, using the Addition or Subtraction Property of Equality.
+ Collect the constants to the other side, using the Addition or Subtraction Property of Equality.
+ Make the coefficient of the variable #math.equation(block: false, alt: "1 ,")[$1 ,$] using the Multiplication or Division Property of Equality.
+ Check the solution by substituting it into the original equation.
]

It is a good idea to make the variable side the one in which the variable has the larger coefficient. This usually makes the arithmetic easier.

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

#solutionbox[
We have #math.equation(block: false, alt: "6 n")[$6 n$] on the left and #math.equation(block: false, alt: "−3 n")[$−3 n$] on the right. Since #math.equation(block: false, alt: "6 greater than minus 3 ,")[$6 > − 3 ,$] make the left side the “variable” side.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed, showing 6n - 2 = -3n + 7. It is a linear equation with one variable, 'n', on both sides of the equals sign.], alt: "A mathematical equation is displayed, showing 6n - 2 = -3n + 7. It is a linear equation with one variable, 'n', on both sides of the equals sign.", caption: none)]),
  [We don't want variables on the right side—add #math.equation(block: false, alt: "3 n")[$3 n$] to both sides to leave only constants on the right.], [#figure(figph[An algebraic equation 6n + 3n - 2 = -3n + 3n + 7, with variables and constants. Some terms are highlighted in red.], alt: "An algebraic equation 6n + 3n - 2 = -3n + 3n + 7, with variables and constants. Some terms are highlighted in red.", caption: none)],
  [Combine like terms.], [#figure(figph[The image displays the algebraic equation 9n - 2 = 7, written in a clear, dark font against a white background.], alt: "The image displays the algebraic equation 9n - 2 = 7, written in a clear, dark font against a white background.", caption: none)],
  [We don't want any constants on the left side, so add 2 to both sides.], [#figure(figph[A mathematical equation shows '9n - 2 + 2 = 7 + 2', where the '+ 2' on both sides of the equals sign is highlighted in red, demonstrating the addition property of equality.], alt: "A mathematical equation shows '9n - 2 + 2 = 7 + 2', where the '+ 2' on both sides of the equals sign is highlighted in red, demonstrating the addition property of equality.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation shows '9n = 9' in black text against a white background.], alt: "A mathematical equation shows '9n = 9' in black text against a white background.", caption: none)],
  [The variable term is on the left and the constant term is on the right. #linebreak() 
   To get the coefficient of #math.equation(block: false, alt: "n")[$n$] to be one, divide both sides by 9.], [#figure(figph[A mathematical equation showing 9n/9 = 9/9, with the number 9 in the denominators highlighted in red.], alt: "A mathematical equation showing 9n/9 = 9/9, with the number 9 in the denominators highlighted in red.", caption: none)],
  [Simplify.], [#figure(figph[The mathematical equation 'n = 1' is shown in a black serif font on a plain white background, symbolizing a foundational value or a starting point in a sequence.], alt: "The mathematical equation 'n = 1' is shown in a black serif font on a plain white background, symbolizing a foundational value or a starting point in a sequence.", caption: none)],
  [Check: Substitute 1 for #math.equation(block: false, alt: "n")[$n$].], [],
  [#figure(figph[The solution to the equation 6n - 2 = -3n + 7 is verified by substituting n=1, resulting in 4 = 4, confirming its correctness.], alt: "The solution to the equation 6n - 2 = -3n + 7 is verified by substituting n=1, resulting in 4 = 4, confirming its correctness.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "8 q minus 5 equals −4 q plus 7 .")[$8 q − 5 = −4 q + 7 .$]

#solutionbox[
#emph[q] = 1
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "7 n minus 3 equals n plus 3 .")[$7 n − 3 = n + 3 .$]

#solutionbox[
#emph[n] = 1
]
]

#examplebox("Example 8")[][
Solve: #math.equation(block: false, alt: "2 a minus 7 equals 5 a plus 8 .")[$2 a − 7 = 5 a + 8 .$]

#solutionbox[
This equation has #math.equation(block: false, alt: "2 a")[$2 a$] on the left and #math.equation(block: false, alt: "5 a")[$5 a$] on the right. Since #math.equation(block: false, alt: "5 greater than 2 ,")[$5 > 2 ,$] make the right side the variable side and the left side the constant side.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation displays '2a - 7 = 5a + 8' in a bold, black serif font on a plain white background.], alt: "A mathematical equation displays '2a - 7 = 5a + 8' in a bold, black serif font on a plain white background.", caption: none)]),
  [Subtract #math.equation(block: false, alt: "2 a")[$2 a$] from both sides to remove the variable term from the left.], [#figure(figph[A math problem demonstrating simplification: 2a - 2a - 7 = 5a - 2a + 8. The red '2a' terms on each side are being subtracted, indicating their removal.], alt: "A math problem demonstrating simplification: 2a - 2a - 7 = 5a - 2a + 8. The red '2a' terms on each side are being subtracted, indicating their removal.", caption: none)],
  [Combine like terms.], [#figure(figph[A mathematical equation is displayed, reading '-7 = 3a + 8' in black text against a white background.], alt: "A mathematical equation is displayed, reading '-7 = 3a + 8' in black text against a white background.", caption: none)],
  [Subtract 8 from both sides to remove the constant from the right.], [#figure(figph[An algebraic equation: -7 - 8 = 3a + 8 - 8. The '8's subtracted from each side are highlighted in red, showing a step to simplify the equation and solve for the variable 'a'.], alt: "An algebraic equation: -7 - 8 = 3a + 8 - 8. The '8's subtracted from each side are highlighted in red, showing a step to simplify the equation and solve for the variable 'a'.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation shows '-15 = 3a' on a white background.], alt: "A mathematical equation shows '-15 = 3a' on a white background.", caption: none)],
  [Divide both sides by 3 to make 1 the coefficient of #math.equation(block: false, alt: "a")[$a$].], [#figure(figph[A mathematical equation is shown with fractions. On the left, -15 is divided by 3, with the 3 in red. On the right, 3a is divided by 3, with the 3 in red. An equals sign separates the two fractions.], alt: "A mathematical equation is shown with fractions. On the left, -15 is divided by 3, with the 3 in red. On the right, 3a is divided by 3, with the 3 in red. An equals sign separates the two fractions.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation is displayed on a white background, reading '-5 = a' in black text, indicating that the variable 'a' is equal to negative five.], alt: "A mathematical equation is displayed on a white background, reading '-5 = a' in black text, indicating that the variable 'a' is equal to negative five.", caption: none)],
  [Check: Let #math.equation(block: false, alt: "a equals −5")[$a = −5$].], [],
  [#figure(figph[Step-by-step verification that a = -5 is the correct solution for the equation 2a - 7 = 5a + 8, resulting in -17 = -17.], alt: "Step-by-step verification that a = -5 is the correct solution for the equation 2a - 7 = 5a + 8, resulting in -17 = -17.", caption: none)], [],
))

Note that we could have made the left side the variable side instead of the right side, but it would have led to a negative coefficient on the variable term. While we could work with the negative, there is less chance of error when working with positives. The strategy outlined above helps avoid the negatives!
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "2 a minus 2 equals 6 a plus 18 .")[$2 a − 2 = 6 a + 18 .$]

#solutionbox[
#emph[a] = −5
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "4 k minus 1 equals 7 k plus 17 .")[$4 k − 1 = 7 k + 17 .$]

#solutionbox[
#emph[k] = −6
]
]

To solve an equation with fractions, we still follow the same steps to get the #strong[solution].

#examplebox("Example 9")[][
Solve: #math.equation(block: false, alt: "the fraction 3 over 2 x plus 5 equals the fraction 1 over 2 x minus 3 .")[$frac(3, 2) #h(0.1em) x + 5 = frac(1, 2) #h(0.1em) x − 3 .$]

#solutionbox[
Since #math.equation(block: false, alt: "the fraction 3 over 2 greater than the fraction 1 over 2 ,")[$frac(3, 2) > frac(1, 2) ,$] make the left side the variable side and the right side the constant side.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed: three-halves x plus five equals one-half x minus three.], alt: "A mathematical equation is displayed: three-halves x plus five equals one-half x minus three.", caption: none)]),
  [Subtract #math.equation(block: false, alt: "the fraction 1 over 2 x")[$frac(1, 2) x$] from both sides.], [#figure(figph[A mathematical equation displayed as 3/2x - 1/2x + 5 = 1/2x - 1/2x - 3, featuring fractional coefficients, the variable 'x', and integer constants, with some terms highlighted in red.], alt: "A mathematical equation displayed as 3/2x - 1/2x + 5 = 1/2x - 1/2x - 3, featuring fractional coefficients, the variable 'x', and integer constants, with some terms highlighted in red.", caption: none)],
  [Combine like terms.], [#figure(figph[A mathematical equation is displayed on a white background, reading 'x + 5 = -3'.], alt: "A mathematical equation is displayed on a white background, reading 'x + 5 = -3'.", caption: none)],
  [Subtract 5 from both sides.], [#figure(figph[A mathematical equation is displayed on a white background: x + 5 - 5 = -3 - 5. The two '5's on the left side are in black, while the '-5' after the 'x+5' and the '-5' after the '-3' are in red.], alt: "A mathematical equation is displayed on a white background: x + 5 - 5 = -3 - 5. The two '5's on the left side are in black, while the '-5' after the 'x+5' and the '-5' after the '-3' are in red.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression on a white background, displaying the equation 'x = -8' in black serif font.], alt: "A mathematical expression on a white background, displaying the equation 'x = -8' in black serif font.", caption: none)],
  [Check: Let #math.equation(block: false, alt: "x equals −8")[$x = −8$].], [],
  [#figure(figph[Verification of a linear equation where x = -8 is substituted into (3/2)x + 5 = (1/2)x - 3, demonstrating that both sides simplify to -7, thereby confirming the solution.], alt: "Verification of a linear equation where x = -8 is substituted into (3/2)x + 5 = (1/2)x - 3, demonstrating that both sides simplify to -7, thereby confirming the solution.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the fraction 7 over 8 x minus 12 equals minus the fraction 1 over 8 x minus 2 .")[$frac(7, 8) #h(0.1em) x − 12 = − frac(1, 8) #h(0.1em) x − 2 .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the fraction 7 over 6 y plus 11 equals the fraction 1 over 6 y plus 8 .")[$frac(7, 6) #h(0.1em) y + 11 = frac(1, 6) #h(0.1em) y + 8 .$]

#solutionbox[
#emph[y] = −3
]
]

We follow the same steps when the equation has decimals, too.

#examplebox("Example 10")[][
Solve: #math.equation(block: false, alt: "3.4 x plus 4 equals 1.6 x minus 5 .")[$3.4 x + 4 = 1.6 x − 5 .$]

#solutionbox[
Since #math.equation(block: false, alt: "3.4 greater than 1.6 ,")[$3.4 > 1.6 ,$] make the left side the variable side and the right side the constant side.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[An algebraic equation is shown on a white background: 3.4x + 4 = 1.6x - 5. The equation involves decimal coefficients, a variable 'x', addition, and subtraction, presented in a standard mathematical format.], alt: "An algebraic equation is shown on a white background: 3.4x + 4 = 1.6x - 5. The equation involves decimal coefficients, a variable 'x', addition, and subtraction, presented in a standard mathematical format.", caption: none)]),
  [Subtract #math.equation(block: false, alt: "1.6 x")[$1.6 x$] from both sides.], [#figure(figph[A mathematical equation is displayed: 3.4x - 1.6x + 4 = 1.6x - 1.6x - 5. The terms -1.6x (left side) and 1.6x - 1.6x (right side) are highlighted in red.], alt: "A mathematical equation is displayed: 3.4x - 1.6x + 4 = 1.6x - 1.6x - 5. The terms -1.6x (left side) and 1.6x - 1.6x (right side) are highlighted in red.", caption: none)],
  [Combine like terms.], [#figure(figph[A mathematical equation, 1.8x + 4 = -5, is displayed in black text on a white background.], alt: "A mathematical equation, 1.8x + 4 = -5, is displayed in black text on a white background.", caption: none)],
  [Subtract 4 from both sides.], [#figure(figph[A mathematical equation '1.8x + 4 - 4 = -5 - 4' is displayed, demonstrating a step in solving for 'x' where the subtraction of '4' from both sides is highlighted in red.], alt: "A mathematical equation '1.8x + 4 - 4 = -5 - 4' is displayed, demonstrating a step in solving for 'x' where the subtraction of '4' from both sides is highlighted in red.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation, 1.8x = -9, is displayed in a black font against a white background.], alt: "A mathematical equation, 1.8x = -9, is displayed in a black font against a white background.", caption: none)],
  [Use the Division Property of Equality.], [#figure(figph[A mathematical equation shows both sides being divided by 1.8: '1.8x / 1.8 = -9 / 1.8', with '1.8' in the denominators highlighted in red, indicating a step to solve for x.], alt: "A mathematical equation shows both sides being divided by 1.8: '1.8x / 1.8 = -9 / 1.8', with '1.8' in the denominators highlighted in red, indicating a step to solve for x.", caption: none)],
  [Simplify.], [#figure(figph[The image displays a mathematical equation in black text on a white background, which reads 'x = -5'.], alt: "The image displays a mathematical equation in black text on a white background, which reads 'x = -5'.", caption: none)],
  [Check: Let #math.equation(block: false, alt: "x equals −5")[$x = −5$].], [],
  [#figure(figph[The image shows the verification of the solution for the equation 3.4x + 4 = 1.6x - 5. By substituting x with -5, both sides of the equation simplify to -13, confirming -5 as the correct solution.], alt: "The image shows the verification of the solution for the equation 3.4x + 4 = 1.6x - 5. By substituting x with -5, both sides of the equation simplify to -13, confirming -5 as the correct solution.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "2.8 x plus 12 equals −1.4 x minus 9 .")[$2.8 x + 12 = −1.4 x − 9 .$]

#solutionbox[
#emph[x] = −5
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "3.6 y plus 8 equals 1.2 y minus 4 .")[$3.6 y + 8 = 1.2 y − 4 .$]

#solutionbox[
#emph[y] = −5
]
]


=== Solve Equations Using a General Strategy

Each of the first few sections of this chapter has dealt with solving one specific form of a #strong[linear equation]. It’s time now to lay out an overall strategy that can be used to solve #emph[any] linear equation. We call this the #emph[general strategy]. Some equations won’t require all the steps to solve, but many will. Simplifying each side of the equation as much as possible first makes the rest of the steps easier.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Use a general strategy for solving linear equations.]

+ Simplify each side of the equation as much as possible. Use the Distributive Property to remove any parentheses. Combine like terms.
+ Collect all the variable terms to one side of the equation. Use the Addition or Subtraction Property of Equality.
+ Collect all the constant terms to the other side of the equation. Use the Addition or Subtraction Property of Equality.
+ Make the coefficient of the variable term to equal to #math.equation(block: false, alt: "1 .")[$1 .$] Use the Multiplication or Division Property of Equality. State the solution to the equation.
+ Check the solution. Substitute the solution into the original equation to make sure the result is a true statement.
]

#examplebox("Example 11")[][
Solve: #math.equation(block: false, alt: "3 open parenthesis x plus 2 close parenthesis equals 18 .")[$3 ( x + 2 ) = 18 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The algebraic equation 3(x + 2) = 18 is displayed on a white background.], alt: "The algebraic equation 3(x + 2) = 18 is displayed on a white background.", caption: none)]),
  [Simplify each side of the equation as much as possible. #linebreak() 
   Use the Distributive Property.], [#figure(figph[A mathematical equation reads '3x + 6 = 18' on a white background. It represents a simple linear equation where the variable 'x' needs to be solved.], alt: "A mathematical equation reads '3x + 6 = 18' on a white background. It represents a simple linear equation where the variable 'x' needs to be solved.", caption: none)],
  [Collect all variable terms on one side of the equation—all #math.equation(block: false, alt: "x")[$x$]s are already on the left side.], [],
  [Collect constant terms on the other side of the equation. #linebreak() 
   Subtract 6 from each side], [#figure(figph[An algebraic equation showing the subtraction of 6 from both sides to solve for x: 3x + 6 - 6 = 18 - 6.], alt: "An algebraic equation showing the subtraction of 6 from both sides to solve for x: 3x + 6 - 6 = 18 - 6.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation, 3x = 12, is displayed in bold black text on a white background.], alt: "A mathematical equation, 3x = 12, is displayed in bold black text on a white background.", caption: none)],
  [Make the coefficient of the variable term equal to 1. Divide each side by 3.], [#figure(figph[A mathematical equation showing 3x divided by 3 equals 12 divided by 3, with the denominator '3' highlighted in red, representing a step in solving for x.], alt: "A mathematical equation showing 3x divided by 3 equals 12 divided by 3, with the denominator '3' highlighted in red, representing a step in solving for x.", caption: none)],
  [Simplify.], [#figure(figph[The equation x = 4 is displayed in black text on a white background.], alt: "The equation x = 4 is displayed in black text on a white background.", caption: none)],
  [Check: Let #math.equation(block: false, alt: "x equals 4")[$x = 4$].], [],
  [#figure(figph[A step-by-step verification of the equation 3(x+2)=18. Substituting x=4 into the equation results in 3(4+2)=18, which simplifies to 3(6)=18, and finally 18=18, confirming x=4 as the solution.], alt: "A step-by-step verification of the equation 3(x+2)=18. Substituting x=4 into the equation results in 3(4+2)=18, which simplifies to 3(6)=18, and finally 18=18, confirming x=4 as the solution.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "5 open parenthesis x plus 3 close parenthesis equals 35 .")[$5 ( x + 3 ) = 35 .$]

#solutionbox[
#emph[x] = 4
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "6 open parenthesis y minus 4 close parenthesis equals −18 .")[$6 ( y − 4 ) = −18 .$]

#solutionbox[
#emph[y] = 1
]
]

#examplebox("Example 12")[][
Solve: #math.equation(block: false, alt: "minus open parenthesis x plus 5 close parenthesis equals 7 .")[$− ( x + 5 ) = 7 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed on a white background: -(x + 5) = 7. The equation shows a negative sign outside parentheses, which enclose the sum of x and 5, equated to the number 7.], alt: "A mathematical equation is displayed on a white background: -(x + 5) = 7. The equation shows a negative sign outside parentheses, which enclose the sum of x and 5, equated to the number 7.", caption: none)]),
  [Simplify each side of the equation as much as possible by distributing. #linebreak() 
   The only #math.equation(block: false, alt: "x")[$x$] term is on the left side, so all variable terms are on the left side of the equation.], [#figure(figph[A mathematical equation is displayed on a white background. The equation reads: -x - 5 = 7.], alt: "A mathematical equation is displayed on a white background. The equation reads: -x - 5 = 7.", caption: none)],
  [Add 5 to both sides to get all constant terms on the right side of the equation.], [#figure(figph[A mathematical equation shows '-x - 5 + 5 = 7 + 5' with the addition of 5 on both sides highlighted in red, demonstrating the first step in solving for x by isolating the variable.], alt: "A mathematical equation shows '-x - 5 + 5 = 7 + 5' with the addition of 5 on both sides highlighted in red, demonstrating the first step in solving for x by isolating the variable.", caption: none)],
  [Simplify.], [#figure(figph[The image displays a mathematical equation: -x = 12. It's a simple algebraic expression where a negative variable 'x' is set equal to the number 12, prompting for the solution of 'x'.], alt: "The image displays a mathematical equation: -x = 12. It's a simple algebraic expression where a negative variable 'x' is set equal to the number 12, prompting for the solution of 'x'.", caption: none)],
  [Make the coefficient of the variable term equal to 1 by multiplying both sides by -1.], [#figure(figph[A mathematical equation showing -1 multiplied by -x equals -1 multiplied by 12, or -1(-x) = -1(12), with the -1 highlighted in red on both sides.], alt: "A mathematical equation showing -1 multiplied by -x equals -1 multiplied by 12, or -1(-x) = -1(12), with the -1 highlighted in red on both sides.", caption: none)],
  [Simplify.], [#figure(figph[The image displays a mathematical equation written in a black font on a white background, stating 'x = -12'.], alt: "The image displays a mathematical equation written in a black font on a white background, stating 'x = -12'.", caption: none)],
  [Check: Let #math.equation(block: false, alt: "x equals −12")[$x = −12$].], [],
  [#figure(figph[A mathematical equation shown as -(x + 5) = 7.], alt: "A mathematical equation shown as -(x + 5) = 7.", caption: none)

 #linebreak() #figure(figph[A mathematical equation reads 'minus open parenthesis minus twelve plus five close parenthesis equals question mark seven' to check if the statement is true.], alt: "A mathematical equation reads 'minus open parenthesis minus twelve plus five close parenthesis equals question mark seven' to check if the statement is true.", caption: none)

 #linebreak() #figure(figph[A mathematical expression showing a question: -(-7) =? 7. This checks understanding of negative numbers and double negatives, where -(-7) simplifies to 7, making the statement true.], alt: "A mathematical expression showing a question: -(-7) =? 7. This checks understanding of negative numbers and double negatives, where -(-7) simplifies to 7, making the statement true.", caption: none)

 #linebreak() #figure(figph[The equation '7=7' is displayed, followed by a checkmark, signifying that the statement is correct and validated.], alt: "The equation '7=7' is displayed, followed by a checkmark, signifying that the statement is correct and validated.", caption: none)

 #linebreak()], [],
))
]
]

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

#solutionbox[
#emph[y] = −6
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "minus open parenthesis z plus 4 close parenthesis equals −12 .")[$− ( z + 4 ) = −12 .$]

#solutionbox[
#emph[z] = 8
]
]

#examplebox("Example 13")[][
Solve: #math.equation(block: false, alt: "4 open parenthesis x minus 2 close parenthesis plus 5 equals −3 .")[$4 ( x − 2 ) + 5 = −3 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed on a white background: 4(x - 2) + 5 = -3.], alt: "A mathematical equation is displayed on a white background: 4(x - 2) + 5 = -3.", caption: none)]),
  [Simplify each side of the equation as much as possible. #linebreak() 
    Distribute.], [#figure(figph[The image displays a mathematical equation: 4x - 8 + 5 = -3. It is an algebraic equation where 'x' is the unknown variable, and the equation needs to be solved to find the value of 'x'.], alt: "The image displays a mathematical equation: 4x - 8 + 5 = -3. It is an algebraic equation where 'x' is the unknown variable, and the equation needs to be solved to find the value of 'x'.", caption: none)],
  [Combine like terms], [#figure(figph[A mathematical equation is displayed on a white background: 4x - 3 = -3.], alt: "A mathematical equation is displayed on a white background: 4x - 3 = -3.", caption: none)],
  [The only #math.equation(block: false, alt: "x")[$x$] is on the left side, so all variable terms are on one side of the equation.], [],
  [Add 3 to both sides to get all constant terms on the other side of the equation.], [#figure(figph[A mathematical equation shows '4x - 3 + 3 = -3 + 3'. The red '+ 3' on both sides indicates the addition of 3 to balance the equation and isolate '4x'.], alt: "A mathematical equation shows '4x - 3 + 3 = -3 + 3'. The red '+ 3' on both sides indicates the addition of 3 to balance the equation and isolate '4x'.", caption: none)],
  [Simplify.], [#figure(figph[The equation 4x = 0 is displayed on a white background, representing a simple algebraic problem.], alt: "The equation 4x = 0 is displayed on a white background, representing a simple algebraic problem.", caption: none)],
  [Make the coefficient of the variable term equal to 1 by dividing both sides by 4.], [#figure(figph[The equation 4x/4 = 0/4 is displayed, with the number 4 in the denominators highlighted in red.], alt: "The equation 4x/4 = 0/4 is displayed, with the number 4 in the denominators highlighted in red.", caption: none)],
  [Simplify.], [#figure(figph[The mathematical equation x=0 is displayed in black text on a white background.], alt: "The mathematical equation x=0 is displayed in black text on a white background.", caption: none)],
  [Check: Let #math.equation(block: false, alt: "x equals 0")[$x = 0$].], [],
  [#figure(figph[Step-by-step solution verification for 4(x-2)+5=-3, showing that x=0 is the correct answer. Substituting x=0 simplifies the equation to -3=-3, confirmed by a checkmark.], alt: "Step-by-step solution verification for 4(x-2)+5=-3, showing that x=0 is the correct answer. Substituting x=0 simplifies the equation to -3=-3, confirmed by a checkmark.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "2 open parenthesis a minus 4 close parenthesis plus 3 equals −1 .")[$2 ( a − 4 ) + 3 = −1 .$]

#solutionbox[
#emph[a] = 2
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "7 open parenthesis n minus 3 close parenthesis minus 8 equals −15 .")[$7 ( n − 3 ) − 8 = −15 .$]

#solutionbox[
#emph[n] = 2
]
]

#examplebox("Example 14")[][
Solve: #math.equation(block: false, alt: "8 minus 2 open parenthesis 3 y plus 5 close parenthesis equals 0 .")[$8 − 2 ( 3 y + 5 ) = 0 .$]

#solutionbox[
Be careful when distributing the negative.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed: 8 - 2(3y + 5) = 0. The equation is rendered in black characters against a plain white background.], alt: "A mathematical equation is displayed: 8 - 2(3y + 5) = 0. The equation is rendered in black characters against a plain white background.", caption: none)]),
  [Simplify—use the Distributive Property.], [#figure(figph[A mathematical equation is displayed, reading '8 - 6y - 10 = 0'.], alt: "A mathematical equation is displayed, reading '8 - 6y - 10 = 0'.", caption: none)],
  [Combine like terms.], [#figure(figph[The image shows a mathematical equation on a white background: -6y - 2 = 0. The equation is displayed in black text.], alt: "The image shows a mathematical equation on a white background: -6y - 2 = 0. The equation is displayed in black text.", caption: none)],
  [Add 2 to both sides to collect constants on the right.], [#figure(figph[A mathematical equation shows '-6y - 2 + 2 = 0 + 2'. The numbers 2 on both sides of the equation are highlighted in red, indicating an operation or a step in solving the equation.], alt: "A mathematical equation shows '-6y - 2 + 2 = 0 + 2'. The numbers 2 on both sides of the equation are highlighted in red, indicating an operation or a step in solving the equation.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation shows '-6y = 2' written in black text on a white background.], alt: "A mathematical equation shows '-6y = 2' written in black text on a white background.", caption: none)],
  [Divide both sides by −6.], [#figure(figph[The equation -6y/-6 = 2/-6 is presented, illustrating division by a negative number in red.], alt: "The equation -6y/-6 = 2/-6 is presented, illustrating division by a negative number in red.", caption: none)],
  [Simplify.], [#figure(figph[The image displays the mathematical equation y = -1/3 in black text on a white background.], alt: "The image displays the mathematical equation y = -1/3 in black text on a white background.", caption: none)],
  [Check: Let #math.equation(block: false, alt: "y equals minus the fraction 1 over 3")[$y = − frac(1, 3)$].], [],
  [#figure(figph[This image demonstrates the verification process for the solution y = -1/3 in the equation 8 - 2(3y + 5) = 0, concluding with 0 = 0, confirming its accuracy.], alt: "This image demonstrates the verification process for the solution y = -1/3 in the equation 8 - 2(3y + 5) = 0, concluding with 0 = 0, confirming its accuracy.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "12 minus 3 open parenthesis 4 j plus 3 close parenthesis equals −17 .")[$12 − 3 ( 4 j + 3 ) = −17 .$]

#solutionbox[
#math.equation(block: true, alt: "j equals the fraction 5 over 3")[$j = frac(5, 3)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "−6 minus 8 open parenthesis k minus 2 close parenthesis equals −10 .")[$−6 − 8 ( k − 2 ) = −10 .$]

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

#examplebox("Example 15")[][
Solve: #math.equation(block: false, alt: "3 open parenthesis x minus 2 close parenthesis minus 5 equals 4 open parenthesis 2 x plus 1 close parenthesis plus 5 .")[$3 ( x − 2 ) − 5 = 4 ( 2 x + 1 ) + 5 .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed, showing 3 multiplied by the quantity x minus 2, then minus 5, which equals 4 multiplied by the quantity 2x plus 1, plus 5.], alt: "A mathematical equation is displayed, showing 3 multiplied by the quantity x minus 2, then minus 5, which equals 4 multiplied by the quantity 2x plus 1, plus 5.", caption: none)]),
  [Distribute.], [#figure(figph[A mathematical equation is displayed on a white background: 3x - 6 - 5 = 8x + 4 + 5.], alt: "A mathematical equation is displayed on a white background: 3x - 6 - 5 = 8x + 4 + 5.", caption: none)],
  [Combine like terms.], [#figure(figph[A mathematical equation is displayed, reading 3x minus 11 equals 8x plus 9, set against a plain white background.], alt: "A mathematical equation is displayed, reading 3x minus 11 equals 8x plus 9, set against a plain white background.", caption: none)],
  [Subtract #math.equation(block: false, alt: "3 x")[$3 x$] to get all the variables on the right since #math.equation(block: false, alt: "8 greater than 3")[$8 > 3$].], [#figure(figph[An algebraic equation is shown: 3x - 3x - 11 = 8x - 3x + 9. The '3x' terms are highlighted in red on both sides of the equation, indicating terms that can be combined or cancelled.], alt: "An algebraic equation is shown: 3x - 3x - 11 = 8x - 3x + 9. The '3x' terms are highlighted in red on both sides of the equation, indicating terms that can be combined or cancelled.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation is displayed on a white background, reading '-11 = 5x + 9'.], alt: "A mathematical equation is displayed on a white background, reading '-11 = 5x + 9'.", caption: none)],
  [Subtract 9 to get the constants on the left.], [#figure(figph[A mathematical equation showing the step to isolate a variable, where '-11 - 9 = 5x + 9 - 9' is displayed with the number 9 highlighted in red on both sides of the equation.], alt: "A mathematical equation showing the step to isolate a variable, where '-11 - 9 = 5x + 9 - 9' is displayed with the number 9 highlighted in red on both sides of the equation.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation is displayed, stating '-20 = 5x' in a clear, dark font against a white background.], alt: "A mathematical equation is displayed, stating '-20 = 5x' in a clear, dark font against a white background.", caption: none)],
  [Divide by 5.], [#figure(figph[A mathematical equation shows 'negative 20 over 5' equals '5x over 5'. The number 5 in the denominators is highlighted in red.], alt: "A mathematical equation shows 'negative 20 over 5' equals '5x over 5'. The number 5 in the denominators is highlighted in red.", caption: none)],
  [Simplify.], [#figure(figph[The mathematical equation '-4 = x' is displayed in black font against a white background.], alt: "The mathematical equation '-4 = x' is displayed in black font against a white background.", caption: none)],
  [Check: Substitute: #math.equation(block: false, alt: "−4 equals x")[$−4 = x$].], [],
  [#figure(figph[Step-by-step verification of an algebraic equation by substituting x = -4 and simplifying both sides to confirm the equality.], alt: "Step-by-step verification of an algebraic equation by substituting x = -4 and simplifying both sides to confirm the equality.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "6 open parenthesis p minus 3 close parenthesis minus 7 equals 5 open parenthesis 4 p plus 3 close parenthesis minus 12 .")[$6 ( p − 3 ) − 7 = 5 ( 4 p + 3 ) − 12 .$]

#solutionbox[
#emph[p] = −2
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "8 open parenthesis q plus 1 close parenthesis minus 5 equals 3 open parenthesis 2 q minus 4 close parenthesis minus 1 .")[$8 ( q + 1 ) − 5 = 3 ( 2 q − 4 ) − 1 .$]

#solutionbox[
#emph[q] = −8
]
]

#examplebox("Example 16")[][
Solve: #math.equation(block: false, alt: "the fraction 1 over 2 open parenthesis 6 x minus 2 close parenthesis equals 5 minus x .")[$frac(1, 2) ( 6 x − 2 ) = 5 − x .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed, showing one-half multiplied by the quantity six-x minus two, which is set equal to five minus x. The equation is written as (1/2)(6x - 2) = 5 - x.], alt: "A mathematical equation is displayed, showing one-half multiplied by the quantity six-x minus two, which is set equal to five minus x. The equation is written as (1/2)(6x - 2) = 5 - x.", caption: none)]),
  [Distribute.], [#figure(figph[A mathematical equation is displayed, reading '3x - 1 = 5 - x' in black text against a white background.], alt: "A mathematical equation is displayed, reading '3x - 1 = 5 - x' in black text against a white background.", caption: none)],
  [Add #math.equation(block: false, alt: "x")[$x$] to get all the variables on the left.], [#figure(figph[A mathematical equation is displayed: 3x - 1 + x = 5 - x + x. On both sides of the equation, the red 'x' terms are added to simplify the expression, demonstrating a step in solving for 'x'.], alt: "A mathematical equation is displayed: 3x - 1 + x = 5 - x + x. On both sides of the equation, the red 'x' terms are added to simplify the expression, demonstrating a step in solving for 'x'.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation, '4x - 1 = 5', is displayed on a white background. The equation is centered in the image.], alt: "A mathematical equation, '4x - 1 = 5', is displayed on a white background. The equation is centered in the image.", caption: none)],
  [Add 1 to get constants on the right.], [#figure(figph[A mathematical equation shows 4x - 1 + 1 = 5 + 1, with the '+ 1' on both sides highlighted in red, illustrating the step of adding the same value to balance the equation and isolate the variable term.], alt: "A mathematical equation shows 4x - 1 + 1 = 5 + 1, with the '+ 1' on both sides highlighted in red, illustrating the step of adding the same value to balance the equation and isolate the variable term.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation is displayed on a white background, reading '4x = 6'.], alt: "A mathematical equation is displayed on a white background, reading '4x = 6'.", caption: none)],
  [Divide by 4.], [#figure(figph[A mathematical equation is displayed on a white background. The equation reads '4x/4 = 6/4', where the '4' in the denominator of both fractions is colored red. The '4x' and '6' are in black.], alt: "A mathematical equation is displayed on a white background. The equation reads '4x/4 = 6/4', where the '4' in the denominator of both fractions is colored red. The '4x' and '6' are in black.", caption: none)],
  [Simplify.], [#figure(figph[The image displays a mathematical equation, x = 3/2, set against a plain white background. The variable 'x' is shown equal to the fraction three halves, with a horizontal line separating the numerator '3' from the denominator '2'.], alt: "The image displays a mathematical equation, x = 3/2, set against a plain white background. The variable 'x' is shown equal to the fraction three halves, with a horizontal line separating the numerator '3' from the denominator '2'.", caption: none)],
  [Check: Let #math.equation(block: false, alt: "x equals the fraction 3 over 2")[$x = frac(3, 2)$].], [],
  [#figure(figph[Step-by-step verification of x=3/2 as the solution for the equation 1/2(6x-2)=5-x, proving equality on both sides.], alt: "Step-by-step verification of x=3/2 as the solution for the equation 1/2(6x-2)=5-x, proving equality on both sides.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the fraction 1 over 3 open parenthesis 6 u plus 3 close parenthesis equals 7 minus u .")[$frac(1, 3) ( 6 u + 3 ) = 7 − u .$]

#solutionbox[
#emph[u] = 2
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the fraction 2 over 3 open parenthesis 9 x minus 12 close parenthesis equals 8 plus 2 x .")[$frac(2, 3) ( 9 x − 12 ) = 8 + 2 x .$]

#solutionbox[
#emph[x] = 4
]
]

In many applications, we will have to solve equations with decimals. The same general strategy will work for these equations.

#examplebox("Example 17")[][
Solve: #math.equation(block: false, alt: "0.24 open parenthesis 100 x plus 5 close parenthesis equals 0.4 open parenthesis 30 x plus 15 close parenthesis .")[$0.24 ( 100 x + 5 ) = 0.4 ( 30 x + 15 ) .$]

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed: 0.24(100x + 5) = 0.4(30x + 15).], alt: "A mathematical equation is displayed: 0.24(100x + 5) = 0.4(30x + 15).", caption: none)]),
  [Distribute.], [#figure(figph[A mathematical equation is displayed, reading '24x + 1.2 = 12x + 6' in a black serif font against a plain white background. It's a linear equation with variables on both sides.], alt: "A mathematical equation is displayed, reading '24x + 1.2 = 12x + 6' in a black serif font against a plain white background. It's a linear equation with variables on both sides.", caption: none)],
  [Subtract #math.equation(block: false, alt: "12 x")[$12 x$] to get all the #math.equation(block: false, alt: "x")[$x$]s to the left.], [#figure(figph[An algebraic equation: 24x + 1.2 - 12x = 12x + 6 - 12x. The term '12x' is highlighted in red, indicating it is being subtracted from both sides to simplify the equation.], alt: "An algebraic equation: 24x + 1.2 - 12x = 12x + 6 - 12x. The term '12x' is highlighted in red, indicating it is being subtracted from both sides to simplify the equation.", caption: none)],
  [Simplify.], [#figure(figph[The image shows the linear equation 12x + 1.2 = 6.], alt: "The image shows the linear equation 12x + 1.2 = 6.", caption: none)],
  [Subtract 1.2 to get the constants to the right.], [#figure(figph[A mathematical equation showing the subtraction of 1.2 from both sides of the equation. The equation reads: 12x + 1.2 - 1.2 = 6 - 1.2. The numbers being subtracted (1.2) are highlighted in red.], alt: "A mathematical equation showing the subtraction of 1.2 from both sides of the equation. The equation reads: 12x + 1.2 - 1.2 = 6 - 1.2. The numbers being subtracted (1.2) are highlighted in red.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation is displayed, showing '12x = 4.8' on a white background.], alt: "A mathematical equation is displayed, showing '12x = 4.8' on a white background.", caption: none)],
  [Divide.], [#figure(figph[An image showing the algebraic equation 12x/12 = 4.8/12, which demonstrates how to isolate the variable x by dividing both sides of the equation by 12. The number 12 is colored red below the fraction bar.], alt: "An image showing the algebraic equation 12x/12 = 4.8/12, which demonstrates how to isolate the variable x by dividing both sides of the equation by 12. The number 12 is colored red below the fraction bar.", caption: none)],
  [Simplify.], [#figure(figph[The mathematical equation x = 0.4 is displayed in black text against a white background.], alt: "The mathematical equation x = 0.4 is displayed in black text against a white background.", caption: none)],
  [Check: Let #math.equation(block: false, alt: "x equals 0.4")[$x = 0.4$].], [],
  [#figure(figph[The image displays the verification of the solution x=0.4 for the equation 0.24(100x+5) = 0.4(30x+15), showing a final equality of 10.8 = 10.8.], alt: "The image displays the verification of the solution x=0.4 for the equation 0.24(100x+5) = 0.4(30x+15), showing a final equality of 10.8 = 10.8.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "0.55 open parenthesis 100 n plus 8 close parenthesis equals 0.6 open parenthesis 85 n plus 14 close parenthesis .")[$0.55 ( 100 n + 8 ) = 0.6 ( 85 n + 14 ) .$]

#solutionbox[
#emph[n] = 1
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "0.15 open parenthesis 40 m minus 120 close parenthesis equals 0.5 open parenthesis 60 m plus 12 close parenthesis .")[$0.15 ( 40 m − 120 ) = 0.5 ( 60 m + 12 ) .$]

#solutionbox[
#emph[m] = −1
]
]

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

- #link("https://www.openstax.org/l/24SolveMultStep")[Solving Multi-Step Equations]
- #link("https://www.openstax.org/l/24SolveEquatVar")[Solve an Equation with Variable Terms on Both Sides]
- #link("https://www.openstax.org/l/24MultiStepEqu")[Solving Multi-Step Equations (L5.4)]
- #link("https://www.openstax.org/l/24EquVarParen")[Solve an Equation with Variables and Parentheses on Both Sides]
]


=== Key Concepts

- #emph[Solve an equation with variables and constants on both sides]+ Choose one side to be the variable side and then the other will be the constant side.
+ Collect the variable terms to the variable side, using the Addition or Subtraction Property of Equality.
+ Collect the constants to the other side, using the Addition or Subtraction Property of Equality.
+ Make the coefficient of the variable 1, using the Multiplication or Division Property of Equality.
+ Check the solution by substituting into the original equation.
- #emph[General strategy for solving linear equations]+ Simplify each side of the equation as much as possible. Use the Distributive Property to remove any parentheses. Combine like terms.
+ Collect all the variable terms to one side of the equation.  Use the Addition or Subtraction Property of Equality.
+ Collect all the constant terms to the other side of the equation. Use the Addition or Subtraction Property of Equality.
+ Make the coefficient of the variable term to equal to 1. Use the Multiplication or Division Property of Equality. State the solution to the equation.
+ Check the solution. Substitute the solution into the original equation to make sure the result is a true statement.


==== Practice Makes Perfect

#strong[Solve an Equation with Constants on Both Sides]

In the following exercises, solve the equation for the variable.

#math.equation(block: true, alt: "6 x minus 2 equals 40")[$6 x − 2 = 40$]

#math.equation(block: true, alt: "7 x minus 8 equals 34")[$7 x − 8 = 34$]

#solutionbox[
#emph[x] = 6
]

#math.equation(block: true, alt: "11 w plus 6 equals 93")[$11 w + 6 = 93$]

#math.equation(block: true, alt: "14 y plus 7 equals 91")[$14 y + 7 = 91$]

#solutionbox[
#emph[y] = 6
]

#math.equation(block: true, alt: "3 a plus 8 equals −46")[$3 a + 8 = −46$]

#math.equation(block: true, alt: "4 m plus 9 equals −23")[$4 m + 9 = −23$]

#solutionbox[
#emph[m] = −8
]

#math.equation(block: true, alt: "−50 equals 7 n minus 1")[$−50 = 7 n − 1$]

#math.equation(block: true, alt: "−47 equals 6 b plus 1")[$−47 = 6 b + 1$]

#solutionbox[
#emph[b] = −8
]

#math.equation(block: true, alt: "25 equals −9 y plus 7")[$25 = −9 y + 7$]

#math.equation(block: true, alt: "29 equals −8 x minus 3")[$29 = −8 x − 3$]

#solutionbox[
#emph[x] = −4
]

#math.equation(block: true, alt: "−12 p minus 3 equals 15")[$−12 p − 3 = 15$]

#math.equation(block: true, alt: "−14 q minus 15 equals 13")[$−14 "q" − 15 = 13$]

#solutionbox[
#emph[q] = −2
]

#strong[Solve an Equation with Variables on Both Sides]

In the following exercises, solve the equation for the variable.

#math.equation(block: true, alt: "8 z equals 7 z minus 7")[$8 z = 7 z − 7$]

#math.equation(block: true, alt: "9 k equals 8 k minus 11")[$9 k = 8 k − 11$]

#solutionbox[
#emph[k] = −11
]

#math.equation(block: true, alt: "4 x plus 36 equals 10 x")[$4 x + 36 = 10 x$]

#math.equation(block: true, alt: "6 x plus 27 equals 9 x")[$6 x + 27 = 9 x$]

#solutionbox[
#emph[x] = 9
]

#math.equation(block: true, alt: "c equals −3 c minus 20")[$c = −3 c − 20$]

#math.equation(block: true, alt: "b equals −4 b minus 15")[$b = −4 b − 15$]

#solutionbox[
#emph[b] = −3
]

#math.equation(block: true, alt: "5 q equals 44 minus 6 q")[$5 q = 44 − 6 q$]

#math.equation(block: true, alt: "7 z equals 39 minus 6 z")[$7 z = 39 − 6 z$]

#solutionbox[
#emph[z] = 3
]

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

#math.equation(block: true, alt: "8 x plus the fraction 3 over 4 equals 7 x")[$8 x + frac(3, 4) = 7 x$]

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

#math.equation(block: true, alt: "−12 a minus 8 equals −16 a")[$−12 a − 8 = −16 a$]

#math.equation(block: true, alt: "−15 r minus 8 equals −11 r")[$−15 r − 8 = −11 r$]

#solutionbox[
#emph[r] = −2
]

#strong[Solve an Equation with Variables and Constants on Both Sides]

In the following exercises, solve the equations for the variable.

#math.equation(block: true, alt: "6 x minus 15 equals 5 x plus 3")[$6 x − 15 = 5 x + 3$]

#math.equation(block: true, alt: "4 x minus 17 equals 3 x plus 2")[$4 x − 17 = 3 x + 2$]

#solutionbox[
#emph[x] = 19
]

#math.equation(block: true, alt: "26 plus 8 d equals 9 d plus 11")[$26 + 8 d = 9 d + 11$]

#math.equation(block: true, alt: "21 plus 6 f equals 7 f plus 14")[$21 + 6 f = 7 f + 14$]

#solutionbox[
#emph[f] = 7
]

#math.equation(block: true, alt: "3 p minus 1 equals 5 p minus 33")[$3 p − 1 = 5 p − 33$]

#math.equation(block: true, alt: "8 q minus 5 equals 5 q minus 20")[$8 q − 5 = 5 q − 20$]

#solutionbox[
#emph[q] = −5
]

#math.equation(block: true, alt: "4 a plus 5 equals minus a minus 40")[$4 a + 5 = − a − 40$]

#math.equation(block: true, alt: "9 c plus 7 equals −2 c minus 37")[$9 c + 7 = −2 c − 37$]

#solutionbox[
#emph[c] = −4
]

#math.equation(block: true, alt: "8 y minus 30 equals −2 y plus 30")[$8 y − 30 = −2 y + 30$]

#math.equation(block: true, alt: "12 x minus 17 equals −3 x plus 13")[$12 x − 17 = −3 x + 13$]

#solutionbox[
#emph[x] = 2
]

#math.equation(block: true, alt: "2 z minus 4 equals 23 minus z")[$2 "z" − 4 = 23 − "z"$]

#math.equation(block: true, alt: "3 y minus 4 equals 12 minus y")[$3 y − 4 = 12 − y$]

#solutionbox[
#emph[y] = 4
]

#math.equation(block: true, alt: "the fraction 5 over 4 c minus 3 equals the fraction 1 over 4 c minus 16")[$frac(5, 4) #h(0.1em) c − 3 = frac(1, 4) #h(0.1em) c − 16$]

#math.equation(block: true, alt: "the fraction 4 over 3 m minus 7 equals the fraction 1 over 3 m minus 13")[$frac(4, 3) #h(0.1em) m − 7 = frac(1, 3) #h(0.1em) m − 13$]

#solutionbox[
#emph[m] = −6
]

#math.equation(block: true, alt: "8 minus the fraction 2 over 5 q equals the fraction 3 over 5 q plus 6")[$8 − frac(2, 5) #h(0.1em) q = frac(3, 5) #h(0.1em) q + 6$]

#math.equation(block: true, alt: "11 minus the fraction 1 over 4 a equals the fraction 3 over 4 a plus 4")[$11 − frac(1, 4) #h(0.1em) a = frac(3, 4) #h(0.1em) a + 4$]

#solutionbox[
#emph[a] = 7
]

#math.equation(block: true, alt: "the fraction 4 over 3 n plus 9 equals the fraction 1 over 3 n minus 9")[$frac(4, 3) #h(0.1em) n + 9 = frac(1, 3) #h(0.1em) n − 9$]

#math.equation(block: true, alt: "the fraction 5 over 4 a plus 15 equals the fraction 3 over 4 a minus 5")[$frac(5, 4) #h(0.1em) a + 15 = frac(3, 4) #h(0.1em) a − 5$]

#solutionbox[
#emph[a] = −40
]

#math.equation(block: true, alt: "the fraction 1 over 4 y plus 7 equals the fraction 3 over 4 y minus 3")[$frac(1, 4) #h(0.1em) y + 7 = frac(3, 4) #h(0.1em) y − 3$]

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

#solutionbox[
#emph[p] = 15
]

#math.equation(block: true, alt: "14 n plus 8.25 equals 9 n plus 19.60")[$14 n + 8.25 = 9 n + 19.60$]

#math.equation(block: true, alt: "13 z plus 6.45 equals 8 z plus 23.75")[$13 z + 6.45 = 8 z + 23.75$]

#solutionbox[
#emph[z] = 3.46
]

#math.equation(block: true, alt: "2.4 w minus 100 equals 0.8 w plus 28")[$2.4 w − 100 = 0.8 w + 28$]

#math.equation(block: true, alt: "2.7 w minus 80 equals 1.2 w plus 10")[$2.7 w − 80 = 1.2 w + 10$]

#solutionbox[
#emph[w] = 60
]

#math.equation(block: true, alt: "5.6 r plus 13.1 equals 3.5 r plus 57.2")[$5.6 r + 13.1 = 3.5 r + 57.2$]

#math.equation(block: true, alt: "6.6 x minus 18.9 equals 3.4 x plus 54.7")[$6.6 x − 18.9 = 3.4 x + 54.7$]

#solutionbox[
#emph[x] = 23
]

#strong[Solve an Equation Using the General Strategy]

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

#math.equation(block: true, alt: "5 open parenthesis x plus 3 close parenthesis equals 75")[$5 ( x + 3 ) = 75$]

#math.equation(block: true, alt: "4 open parenthesis y plus 7 close parenthesis equals 64")[$4 ( y + 7 ) = 64$]

#solutionbox[
#emph[y] = 9
]

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

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

#solutionbox[
#emph[x] = 6
]

#math.equation(block: true, alt: "20 open parenthesis y minus 8 close parenthesis equals −60")[$20 ( y − 8 ) = −60$]

#math.equation(block: true, alt: "14 open parenthesis y minus 6 close parenthesis equals −42")[$14 ( y − 6 ) = −42$]

#solutionbox[
#emph[y] = 3
]

#math.equation(block: true, alt: "−4 open parenthesis 2 n plus 1 close parenthesis equals 16")[$−4 ( 2 n + 1 ) = 16$]

#math.equation(block: true, alt: "−7 open parenthesis 3 n plus 4 close parenthesis equals 14")[$−7 ( 3 n + 4 ) = 14$]

#solutionbox[
#emph[n] = −2
]

#math.equation(block: true, alt: "3 open parenthesis 10 plus 5 r close parenthesis equals 0")[$3 ( 10 + 5 r ) = 0$]

#math.equation(block: true, alt: "8 open parenthesis 3 plus 3 p close parenthesis equals 0")[$8 ( 3 + 3 "p" ) = 0$]

#solutionbox[
#emph[p] = −1
]

#math.equation(block: true, alt: "the fraction 2 over 3 open parenthesis 9 c minus 3 close parenthesis equals 22")[$frac(2, 3) ( 9 c − 3 ) = 22$]

#math.equation(block: true, alt: "the fraction 3 over 5 open parenthesis 10 x minus 5 close parenthesis equals 27")[$frac(3, 5) ( 10 x − 5 ) = 27$]

#solutionbox[
#emph[x] = 5
]

#math.equation(block: true, alt: "5 open parenthesis 1.2 u minus 4.8 close parenthesis equals −12")[$5 ( 1.2 u − 4.8 ) = −12$]

#math.equation(block: true, alt: "4 open parenthesis 2.5 v minus 0.6 close parenthesis equals 7.6")[$4 ( 2.5 v − 0.6 ) = 7.6$]

#solutionbox[
#emph[v] = 1
]

#math.equation(block: true, alt: "0.2 open parenthesis 30 n plus 50 close parenthesis equals 28")[$0.2 ( 30 n + 50 ) = 28$]

#math.equation(block: true, alt: "0.5 open parenthesis 16 m plus 34 close parenthesis equals −15")[$0.5 ( 16 m + 34 ) = −15$]

#solutionbox[
#emph[m] = 0.25
]

#math.equation(block: true, alt: "minus open parenthesis w minus 6 close parenthesis equals 24")[$− ( w − 6 ) = 24$]

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

#solutionbox[
#emph[t] = −9
]

#math.equation(block: true, alt: "9 open parenthesis 3 a plus 5 close parenthesis plus 9 equals 54")[$9 ( 3 a + 5 ) + 9 = 54$]

#math.equation(block: true, alt: "8 open parenthesis 6 b minus 7 close parenthesis plus 23 equals 63")[$8 ( 6 b − 7 ) + 23 = 63$]

#solutionbox[
#emph[b] = 2
]

#math.equation(block: true, alt: "10 plus 3 open parenthesis z plus 4 close parenthesis equals 19")[$10 + 3 ( z + 4 ) = 19$]

#math.equation(block: true, alt: "13 plus 2 open parenthesis m minus 4 close parenthesis equals 17")[$13 + 2 ( m − 4 ) = 17$]

#solutionbox[
#emph[m] = 6
]

#math.equation(block: true, alt: "7 plus 5 open parenthesis 4 minus q close parenthesis equals 12")[$7 + 5 ( 4 − q ) = 12$]

#math.equation(block: true, alt: "−9 plus 6 open parenthesis 5 minus k close parenthesis equals 12")[$−9 + 6 ( 5 − k ) = 12$]

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

#math.equation(block: true, alt: "15 minus open parenthesis 3 r plus 8 close parenthesis equals 28")[$15 − ( 3 r + 8 ) = 28$]

#math.equation(block: true, alt: "18 minus open parenthesis 9 r plus 7 close parenthesis equals −16")[$18 − ( 9 r + 7 ) = −16$]

#solutionbox[
#emph[r] = 3
]

#math.equation(block: true, alt: "11 minus 4 open parenthesis y minus 8 close parenthesis equals 43")[$11 − 4 ( y − 8 ) = 43$]

#math.equation(block: true, alt: "18 minus 2 open parenthesis y minus 3 close parenthesis equals 32")[$18 − 2 ( y − 3 ) = 32$]

#solutionbox[
#emph[y] = −4
]

#math.equation(block: true, alt: "9 open parenthesis p minus 1 close parenthesis equals 6 open parenthesis 2 p minus 1 close parenthesis")[$9 ( p − 1 ) = 6 ( 2 p − 1 )$]

#math.equation(block: true, alt: "3 open parenthesis 4 n minus 1 close parenthesis minus 2 equals 8 n plus 3")[$3 ( 4 n − 1 ) − 2 = 8 n + 3$]

#solutionbox[
#emph[n] = 2
]

#math.equation(block: true, alt: "9 open parenthesis 2 m minus 3 close parenthesis minus 8 equals 4 m plus 7")[$9 ( 2 m − 3 ) − 8 = 4 m + 7$]

#math.equation(block: true, alt: "5 open parenthesis x minus 4 close parenthesis minus 4 x equals 14")[$5 ( x − 4 ) − 4 x = 14$]

#solutionbox[
#emph[x] = 34
]

#math.equation(block: true, alt: "8 open parenthesis x minus 4 close parenthesis minus 7 x equals 14")[$8 ( x − 4 ) − 7 x = 14$]

#math.equation(block: true, alt: "5 plus 6 open parenthesis 3 s minus 5 close parenthesis equals −3 plus 2 open parenthesis 8 s minus 1 close parenthesis")[$5 + 6 ( 3 s − 5 ) = −3 + 2 ( 8 s − 1 )$]

#solutionbox[
#emph[s] = 10
]

#math.equation(block: true, alt: "−12 plus 8 open parenthesis x minus 5 close parenthesis equals −4 plus 3 open parenthesis 5 x minus 2 close parenthesis")[$−12 + 8 ( x − 5 ) = −4 + 3 ( 5 x − 2 )$]

#math.equation(block: true, alt: "4 open parenthesis x minus 1 close parenthesis minus 8 equals 6 open parenthesis 3 x minus 2 close parenthesis minus 7")[$4 ( x − 1 ) − 8 = 6 ( 3 x − 2 ) − 7$]

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

#math.equation(block: true, alt: "7 open parenthesis 2 x minus 5 close parenthesis equals 8 open parenthesis 4 x minus 1 close parenthesis minus 9")[$7 ( 2 x − 5 ) = 8 ( 4 x − 1 ) − 9$]


==== Everyday Math

#strong[Making a fence] Jovani has a fence around the rectangular garden in his backyard. The perimeter of the fence is #math.equation(block: false, alt: "150")[$150$] feet. The length is #math.equation(block: false, alt: "15")[$15$] feet more than the width. Find the width, #math.equation(block: false, alt: "w ,")[$w ,$] by solving the equation #math.equation(block: false, alt: "150 equals 2 open parenthesis w plus 15 close parenthesis plus 2 w .")[$150 = 2 ( w + 15 ) + 2 w .$]

#solutionbox[
30 feet
]

#strong[Concert tickets] At a school concert, the total value of tickets sold was #math.equation(block: false, alt: "$1,506.")[$"$1,506."$] Student tickets sold for #math.equation(block: false, alt: "$6")[$"$6"$] and adult tickets sold for #math.equation(block: false, alt: "$9.")[$"$9."$] The number of adult tickets sold was #math.equation(block: false, alt: "5")[$5$] less than #math.equation(block: false, alt: "3")[$3$] times the number of student tickets. Find the number of student tickets sold, #math.equation(block: false, alt: "s ,")[$s ,$] by solving the equation #math.equation(block: false, alt: "6 s plus 9 open parenthesis 3 s minus 5 close parenthesis equals 1506 .")[$6 s + 9 ( 3 s − 5 ) = 1506 .$]

#strong[Coins] Rhonda has #math.equation(block: false, alt: "$1.90")[$"$1.90"$] in nickels and dimes. The number of dimes is one less than twice the number of nickels. Find the number of nickels, #math.equation(block: false, alt: "n ,")[$n ,$] by solving the equation #math.equation(block: false, alt: "0.05 n plus 0.10 open parenthesis 2 n minus 1 close parenthesis equals 1.90 .")[$0.05 n + 0.10 ( 2 n − 1 ) = 1.90 .$]

#solutionbox[
8 nickels
]

#strong[Fencing] Micah has #math.equation(block: false, alt: "74")[$74$] feet of fencing to make a rectangular dog pen in his yard. He wants the length to be #math.equation(block: false, alt: "25")[$25$] feet more than the width. Find the length, #math.equation(block: false, alt: "L ,")[$L ,$] by solving the equation #math.equation(block: false, alt: "2 L plus 2 open parenthesis L minus 25 close parenthesis equals 74 .")[$2 L + 2 ( L − 25 ) = 74 .$]


==== Writing Exercises

When solving an equation with variables on both sides, why is it usually better to choose the side with the larger coefficient as the variable side?

#solutionbox[
Answers will vary.
]

Solve the equation #math.equation(block: false, alt: "10 x plus 14 equals −2 x plus 38 ,")[$10 x + 14 = −2 x + 38 ,$] explaining all the steps of your solution.

What is the first step you take when solving the equation #math.equation(block: false, alt: "3 minus 7 open parenthesis y minus 4 close parenthesis equals 38 ?")[$3 − 7 ( y − 4 ) = 38 ?$] Explain why this is your first step.

#solutionbox[
Answers will vary.
]

Solve the equation #math.equation(block: false, alt: "the fraction 1 over 4 open parenthesis 8 x plus 20 close parenthesis equals 3 x minus 4")[$frac(1, 4) ( 8 x + 20 ) = 3 x − 4$] explaining all the steps of your solution as in the examples in this section.

Using your own words, list the steps in the General Strategy for Solving Linear Equations.

#solutionbox[
Answers will vary.
]

Explain why you should simplify both sides of an equation as much as possible before collecting the variable terms to one side and the constant terms to the other side.


==== Self Check

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

#figure(figph[Self-evaluation grid for algebraic equation solving skills, covering equations with constants, variables, or both on both sides, rated by confidence level.], alt: "Self-evaluation grid for algebraic equation solving skills, covering equations with constants, variables, or both on both sides, rated by confidence level.", caption: none)

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?