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== 2.3#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")[$4 y$]
]
]


=== Solve Equations with Constants on Both Sides

In all the equations we have solved so far, all the variable terms were on only one side of the equation with the constants on the other side. This does not happen all the time—so now we will learn to solve equations in which the variable terms, or constant terms, or both 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 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 began 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: "7 x plus 8 equals −13 .")[$7 x + 8 = −13 .$]

#solutionbox[
In this equation, the variable is found 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 will write the labels above the equation to help us remember what goes where.

#figure(figph[This figure shows the equation 7x plus 8 equals negative 13, with the left side of the equation labeled “variable”, written in red, and the right side of the equation labeled “constant”, written in red.], alt: "This figure shows the equation 7x plus 8 equals negative 13, with the left side of the equation labeled “variable”, written in red, and the right side of the equation labeled “constant”, written in red.", caption: none)

Since the left side is the “#math.equation(block: false, alt: "x")[$x$]”, or variable side, the 8 is out of place. We must “undo” adding 8 by subtracting 8, and to keep the equality we must subtract 8 from both sides.

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[Identifying variables and constants in the algebraic equation 7x + 8 = -13.], alt: "Identifying variables and constants in the algebraic equation 7x + 8 = -13.", caption: none)], [],
  [Use the Subtraction Property of Equality.], [#figure(figph[A step in solving the equation 7x + 8 = -13, showing 8 being subtracted from both sides of the equation to isolate the 7x term. The subtracted 8s are highlighted in red.], alt: "A step in solving the equation 7x + 8 = -13, showing 8 being subtracted from both sides of the equation to isolate the 7x term. The subtracted 8s are highlighted in red.", caption: none)], [],
  [Simplify.], [#figure(figph[A mathematical equation is displayed, reading '7x = -21' against a plain white background.], alt: "A mathematical equation is displayed, reading '7x = -21' against a plain white background.", caption: none)], [],
  [Now all the variables are on the left and the constant on the right.  #linebreak() The equation looks like those you learned to solve earlier.], [], [],
  [Use the Division Property of Equality.], [#figure(figph[A step in solving an equation, showing 7x divided by 7 equals -21 divided by 7, with the divisor '7' highlighted in red on both sides.], alt: "A step in solving an equation, showing 7x divided by 7 equals -21 divided by 7, with the divisor '7' highlighted in red on both sides.", caption: none)], [],
  [Simplify.], [#figure(figph[A mathematical equation is shown with 'X = -3' in bold black text on a white background.], alt: "A mathematical equation is shown with 'X = -3' in bold black text on a white background.", caption: none)], [],
  [Check:], [#figure(figph[The image displays the algebraic equation 7x + 8 = -13, which is a linear equation in one variable.], alt: "The image displays the algebraic equation 7x + 8 = -13, which is a linear equation in one variable.", caption: none)], [],
  [Let #math.equation(block: false, alt: "x equals −3")[$x = −3$].], [#figure(figph[A mathematical equation: 7(-3) + 8 =? -13. The left side calculates to -21 + 8 = -13, confirming that -13 indeed equals -13, making the statement true.], alt: "A mathematical equation: 7(-3) + 8 =? -13. The left side calculates to -21 + 8 = -13, confirming that -13 indeed equals -13, making the statement true.", caption: none)], [],
  [], [#figure(figph[A mathematical problem showing -21 + 8 ?= -13, which is solved and verified as -13 = -13 with a checkmark, confirming the equality.], alt: "A mathematical problem showing -21 + 8 ?= -13, which is solved and verified as -13 = -13 with a checkmark, confirming the equality.", 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[
#math.equation(block: true, alt: "x equals −4")[$x = −4$]
]
]

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

#examplebox("Example 2")[][
Solve: #math.equation(block: false, alt: "8 y minus 9 equals 31 .")[$8 y − 9 = 31 .$]

#solutionbox[
Notice, the variable is only on the left side of the equation, so we will call this side the “variable” side, and the right side will be the “constant” side. Since the left side is the “variable” side, the 9 is out of place. It is subtracted from the #math.equation(block: false, alt: "8 y")[$8 y$], so to “undo” subtraction, add 9 to both sides. Remember, whatever you do to the left, you must do to the right.

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[A mathematical equation '8y - 9 = 31' is displayed. Above '8y' is the label 'variable' in red, and above '31' is 'constant' in red, highlighting components of the equation.], alt: "A mathematical equation '8y - 9 = 31' is displayed. Above '8y' is the label 'variable' in red, and above '31' is 'constant' in red, highlighting components of the equation.", caption: none)], [],
  [Add 9 to both sides.], [#figure(figph[A step in solving an algebraic equation, showing 8y - 9 + 9 = 31 + 9. The number 9 is added to both sides of the equation, with the additions highlighted in red.], alt: "A step in solving an algebraic equation, showing 8y - 9 + 9 = 31 + 9. The number 9 is added to both sides of the equation, with the additions highlighted in red.", caption: none)], [],
  [Simplify.], [#figure(figph[The image displays the algebraic equation '8y = 40' in black text against a white background.], alt: "The image displays the algebraic equation '8y = 40' in black text against a white background.", caption: none)], [],
  [], [The variables are now on one side and the constants on the other. #linebreak() We continue from here as we did earlier.], [],
  [Divide both sides by 8.], [#figure(figph[A mathematical equation shows '8y over 8 equals 40 over 8', with the denominators '8' in red, indicating division on both sides to solve for 'y'.], alt: "A mathematical equation shows '8y over 8 equals 40 over 8', with the denominators '8' in red, indicating division on both sides to solve for 'y'.", caption: none)], [],
  [Simplify.], [#figure(figph[The image displays the mathematical equation 'y = 5' in a clear, black font against a white background.], alt: "The image displays the mathematical equation 'y = 5' in a clear, black font against a white background.", caption: none)], [],
  [Check:], [#figure(figph[The image displays the algebraic equation 8y - 9 = 31 in black text against a white background.], alt: "The image displays the algebraic equation 8y - 9 = 31 in black text against a white background.", caption: none)], [],
  [Let #math.equation(block: false, alt: "y equals 5")[$y = 5$].], [#figure(figph[A mathematical equation is displayed: 8 multiplied by 5 minus 9, followed by a question mark and an equals sign, then 31. The number 5 is highlighted in red.], alt: "A mathematical equation is displayed: 8 multiplied by 5 minus 9, followed by a question mark and an equals sign, then 31. The number 5 is highlighted in red.", caption: none)], [],
  [], [#figure(figph[The equation 40 - 9 ?= 31 is displayed, prompting verification of the subtraction.], alt: "The equation 40 - 9 ?= 31 is displayed, prompting verification of the subtraction.", caption: none)], [],
  [], [#figure(figph[The image shows the equation '31 = 31' followed by a checkmark, signifying that the equality is correct and validated.], alt: "The image shows the equation '31 = 31' followed by a checkmark, signifying that the equality is correct and validated.", 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[
#math.equation(block: true, alt: "y equals 5")[$y = 5$]
]
]

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


=== Solve Equations with Variables on Both Sides

What if there are #strong[variables] on both sides of the equation? For equations like this, begin as we did above—choose a “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 constants on the other side.

#examplebox("Example 3")[][
Solve: #math.equation(block: false, alt: "9 x equals 8 x minus 6 .")[$9 x = 8 x − 6 .$]

#solutionbox[
Here the variable is on both sides, but the constants only appear 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 example of an algebraic equation 9x = 8x - 6, with 'variable' and 'constant' terms labeled in red.], alt: "An example of an algebraic equation 9x = 8x - 6, with 'variable' and 'constant' terms labeled in red.", caption: none)], [],
  [We don’t want any #math.equation(block: false, alt: "x")[$x$]’s on the right, so subtract the #math.equation(block: false, alt: "8 x")[$8 x$] from both sides.], [#figure(figph[A mathematical equation shows '9x - 8x = 8x - 8x - 6'. The '8x' terms being subtracted on both sides are highlighted in red, indicating a step in solving the equation.], alt: "A mathematical equation shows '9x - 8x = 8x - 8x - 6'. The '8x' terms being subtracted on both sides are highlighted in red, indicating a step in solving the equation.", caption: none)], [],
  [Simplify.], [#figure(figph[The mathematical equation X = -6 is displayed in the center of a plain white background.], alt: "The mathematical equation X = -6 is displayed in the center of a plain white background.", caption: none)], [],
  [We succeeded in getting the variables on one side and the constants on the other, and have obtained the solution.], [], [],
  [Check:], [#figure(figph[A mathematical equation is displayed against a white background, reading '9x = 8x - 6'.], alt: "A mathematical equation is displayed against a white background, reading '9x = 8x - 6'.", caption: none)], [],
  [Let #math.equation(block: false, alt: "x equals −6")[$x = −6$].], [#figure(figph[An algebra equation showing 9(-6) ?= 8(-6) - 6, prompting verification of equality. The equation is true as both sides equal -54.], alt: "An algebra equation showing 9(-6) ?= 8(-6) - 6, prompting verification of equality. The equation is true as both sides equal -54.", caption: none)], [],
  [], [#figure(figph[A math problem shows '-54 ?= -48 - 6' asking to fill in the correct comparison operator. The right side simplifies to -54, meaning the question mark should be an equals sign.], alt: "A math problem shows '-54 ?= -48 - 6' asking to fill in the correct comparison operator. The right side simplifies to -54, meaning the question mark should be an equals sign.", caption: none)], [],
  [], [#figure(figph[A mathematical equation showing -54 equals -54, with a checkmark indicating correctness.], alt: "A mathematical equation showing -54 equals -54, with a checkmark indicating correctness.", caption: none)], [],
))
]
]

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

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

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

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

#examplebox("Example 4")[][
Solve: #math.equation(block: false, alt: "5 y minus 9 equals 8 y .")[$5 y − 9 = 8 y .$]

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

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[An algebraic equation, 5y - 9 = 8y, is displayed with the terms 'constant' and 'variable' highlighted above, demonstrating basic components of algebra.], alt: "An algebraic equation, 5y - 9 = 8y, is displayed with the terms 'constant' and 'variable' highlighted above, demonstrating basic components of algebra.", caption: none)], [],
  [Subtract #math.equation(block: false, alt: "5 y")[$5 y$] from both sides.], [#figure(figph[A mathematical equation shows 5y - 5y - 9 = 8y - 5y, with the terms -5y highlighted in red on both sides of the equality.], alt: "A mathematical equation shows 5y - 5y - 9 = 8y - 5y, with the terms -5y highlighted in red on both sides of the equality.", caption: none)], [],
  [Simplify.], [#figure(figph[The image displays the algebraic equation -9 = 3y, which can be solved to find the value of y.], alt: "The image displays the algebraic equation -9 = 3y, which can be solved to find the value of y.", caption: none)], [],
  [We have the #emph[y]’s on the right and the #linebreak() constants on the left. Divide both sides by 3.], [#figure(figph[A mathematical equation showing negative nine over three equals three y over three, written in black text with the denominators in red.], alt: "A mathematical equation showing negative nine over three equals three y over three, written in black text with the denominators in red.", caption: none)], [],
  [Simplify.], [#figure(figph[A mathematical equation displays '-3 = y' on a white background.], alt: "A mathematical equation displays '-3 = y' on a white background.", caption: none)], [],
  [Check:], [#figure(figph[The image shows the algebraic equation 5y - 9 = 8y.], alt: "The image shows the algebraic equation 5y - 9 = 8y.", caption: none)], [],
  [Let #math.equation(block: false, alt: "y equals −3 .")[$y = −3 .$]], [#figure(figph[A mathematical equation shows '5(-3) - 9' on the left side, an equals sign with a question mark above it in the middle, and '8(-3)' on the right side. The number -3 is highlighted in red on both sides.], alt: "A mathematical equation shows '5(-3) - 9' on the left side, an equals sign with a question mark above it in the middle, and '8(-3)' on the right side. The number -3 is highlighted in red on both sides.", caption: none)], [],
  [], [#figure(figph[An image displaying the calculation -15 - 9 = -24, with a question mark indicating a verification of the equality.], alt: "An image displaying the calculation -15 - 9 = -24, with a question mark indicating a verification of the equality.", caption: none)], [],
  [], [#figure(figph[A mathematical equation shows '-24 = -24' with a checkmark, indicating the equality is correct. It represents a verified true statement.], alt: "A mathematical equation shows '-24 = -24' with a checkmark, indicating the equality is correct. It represents a verified true statement.", 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[
#math.equation(block: true, alt: "p equals −7")[$p = −7$]
]
]

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

#examplebox("Example 5")[][
Solve: #math.equation(block: false, alt: "12 x equals − x plus 26 .")[$12 x = "−" x + 26 .$]

#solutionbox[
The only constant 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, 12x = -x + 26, is displayed with the terms 'variable' and 'constant' highlighted above their respective parts, demonstrating how to identify these components in an expression.], alt: "An algebraic equation, 12x = -x + 26, is displayed with the terms 'variable' and 'constant' highlighted above their respective parts, demonstrating how to identify these components in an expression.", caption: none)]),
  [Remove the #math.equation(block: false, alt: "− x")[$− x$] from the right side by adding #math.equation(block: false, alt: "x")[$x$] to both sides.], [#figure(figph[A mathematical equation is displayed, showing '12x + x = -x + x + 26' with some plus signs and the variable 'x' highlighted in red, indicating a step in solving or simplifying the equation.], alt: "A mathematical equation is displayed, showing '12x + x = -x + x + 26' with some plus signs and the variable 'x' highlighted in red, indicating a step in solving or simplifying the equation.", caption: none)],
  [Simplify.], [#figure(figph[A simple algebraic equation is displayed, 13x = 26. This equation requires solving for the variable x, where 13 times x equals 26.], alt: "A simple algebraic equation is displayed, 13x = 26. This equation requires solving for the variable x, where 13 times x equals 26.", caption: none)],
  [All the #math.equation(block: false, alt: "x")[$x$]’s are on the left and the constants are on the right. Divide both sides by 13.], [#figure(figph[A mathematical equation showing the division of both sides of an equation by 13. The equation is 13x over 13 equals 26 over 13, with the denominator 13 written in red on both sides.], alt: "A mathematical equation showing the division of both sides of an equation by 13. The equation is 13x over 13 equals 26 over 13, with the denominator 13 written in red on both sides.", caption: none)],
  [Simplify.], [#figure(figph[The equation x = 2 is displayed in black text on a white background.], alt: "The equation x = 2 is displayed in black text on a white background.", 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[
#math.equation(block: true, alt: "j equals 2")[$j = 2$]
]
]

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


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

The next example will be the first to have variables and constants on both sides of the equation. It may take several steps to solve this equation, so we need a clear and organized strategy.

#examplebox("Example 6")[How to Solve Equations with Variables and Constants on Both Sides][
Solve: #math.equation(block: false, alt: "7 x plus 5 equals 6 x plus 2 .")[$7 x + 5 = 6 x + 2 .$]

#solutionbox[
#figure(figph[This figure is a table that has three columns and four rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads: “Step 1. Choose which side will the “variable” side—the other side will be the “constant” side.” The text in the second cell reads: “The variable terms are 7 x and 6 x. Since 7 is greater than 6, we will make the left side the “x” side and so the right side will be the “constant” side.” The third cell contains the equation 7 x plus 5 equals 6 x plus 2, and the left side of the equation is labeled “variable” written in red, and the right side of the equation is labeled “constant” written in red.], alt: "This figure is a table that has three columns and four rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads: “Step 1. Choose which side will the “variable” side—the other side will be the “constant” side.” The text in the second cell reads: “The variable terms are 7 x and 6 x. Since 7 is greater than 6, we will make the left side the “x” side and so the right side will be the “constant” side.” The third cell contains the equation 7 x plus 5 equals 6 x plus 2, and the left side of the equation is labeled “variable” written in red, and the right side of the equation is labeled “constant” written in red.", caption: none)

#figure(figph[In the second row of the table, the first cell says: “Step 2. Collect the variable terms to the “variable” side of the equation, using the addition or subtraction property of equality.” In the second cell, the instructions say: “ With the right side as the “constant” side, the 6x is out of place, so subtract 6x from both sides. Combine like terms. Now the variable is only on the left side!” The third cell contains the original equation with 6x subtracted from both sides: 7 x minus 6 x plus 5 equals 6 x minus 6 x plus 2, with “minus 6 x” written in red on both sides. Below this is the same equation with like terms combined: x plus 5 equals 2.], alt: "In the second row of the table, the first cell says: “Step 2. Collect the variable terms to the “variable” side of the equation, using the addition or subtraction property of equality.” In the second cell, the instructions say: “ With the right side as the “constant” side, the 6x is out of place, so subtract 6x from both sides. Combine like terms. Now the variable is only on the left side!” The third cell contains the original equation with 6x subtracted from both sides: 7 x minus 6 x plus 5 equals 6 x minus 6 x plus 2, with “minus 6 x” written in red on both sides. Below this is the same equation with like terms combined: x plus 5 equals 2.", caption: none)

#figure(figph[In the third row of the table, the first cell says: “Step 3. Collect all the constants to the other side of the equation, using the addition or subtraction property of equality.” In the second cell, the instructions say: “The right side is the “constant” side, so the 5 is out of place. Subtract 5 from both sides. Simplify.” The third cell contains the equation x plus 5 minus 5 equals 2 minus 5, with “minus 5” written in red on both sides. Below this is the answer to the equation: x equals negative 3.], alt: "In the third row of the table, the first cell says: “Step 3. Collect all the constants to the other side of the equation, using the addition or subtraction property of equality.” In the second cell, the instructions say: “The right side is the “constant” side, so the 5 is out of place. Subtract 5 from both sides. Simplify.” The third cell contains the equation x plus 5 minus 5 equals 2 minus 5, with “minus 5” written in red on both sides. Below this is the answer to the equation: x equals negative 3.", caption: none)

#figure(figph[In the fourth row of the table, the first cell says: “Step 4. Make the coefficient of the variable equal 1, using the Multiplication or Division Property of Equality.” In the second cell, the instructions say: “The coefficient of x is one. The equation is solved.” The third cell is blank.], alt: "In the fourth row of the table, the first cell says: “Step 4. Make the coefficient of the variable equal 1, using the Multiplication or Division Property of Equality.” In the second cell, the instructions say: “The coefficient of x is one. The equation is solved.” The third cell is blank.", caption: none)

#figure(figph[In the fifth row of the table, the first cell says: “Step 5. Check.” The instructions in the second cell say: “Check. Let x equal negative 3. Simplify. Add.” In the third cell is the original equation again: 7 x plus 5 equals 6x plus 2. Below this is the same equation with negative 3 substituted in for x: 7 times negative 3 (in paretheses) plus 5 might equal 6 times negative 3 (in parentheses) plus 2, with the “times negative 3” written in red on both sides of the equation. Below this is the equation negative 21 plus 5 might equal negative 18 plus 2. On the last line is the equation negative 16 equals negative 16, with a check mark next to it.], alt: "In the fifth row of the table, the first cell says: “Step 5. Check.” The instructions in the second cell say: “Check. Let x equal negative 3. Simplify. Add.” In the third cell is the original equation again: 7 x plus 5 equals 6x plus 2. Below this is the same equation with negative 3 substituted in for x: 7 times negative 3 (in paretheses) plus 5 might equal 6 times negative 3 (in parentheses) plus 2, with the “times negative 3” written in red on both sides of the equation. Below this is the equation negative 21 plus 5 might equal negative 18 plus 2. On the last line is the equation negative 16 equals negative 16, with a check mark next to it.", 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[
#math.equation(block: true, alt: "x equals −1")[$x = −1$]
]
]

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

We’ll list the steps below so you can easily refer to them. But we’ll call this the ‘Beginning Strategy’ because we’ll be adding some steps later in this chapter.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Beginning Strategy for Solving Equations with Variables and Constants on Both Sides of the Equation.]

+ Choose which side will be the “variable” side—the other side will be the “constant” side.
+ Collect the variable terms to the “variable” side of the equation, using the Addition or Subtraction Property of Equality.
+ Collect all the constants to the other side of the equation, using the Addition or Subtraction Property of Equality.
+ Make the coefficient of the variable equal 1, using the Multiplication or Division Property of Equality.
+ Check the solution by substituting it into the original equation.
]

In Step 1, a helpful approach is to make the “variable” side the side that has the variable with the larger coefficient. This usually makes the arithmetic easier.

#examplebox("Example 7")[][
Solve: #math.equation(block: false, alt: "8 n minus 4 equals −2 n plus 6 .")[$8 n − 4 = −2 n + 6 .$]

#solutionbox[
In the first step, choose the variable side by comparing the coefficients of the variables on each side.

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [Since #math.equation(block: false, alt: "8 greater than −2")[$8 > −2$], make the left side the “variable” side.], [#figure(figph[An algebraic equation, 8n - 4 = -2n + 6, is displayed with the terms labeled. 'Variable' is above 8n and -2n, and 'constant' is above -4 and +6, indicating their mathematical roles.], alt: "An algebraic equation, 8n - 4 = -2n + 6, is displayed with the terms labeled. 'Variable' is above 8n and -2n, and 'constant' is above -4 and +6, indicating their mathematical roles.", caption: none)], [],
  [We don’t want variable terms on the right side—add #math.equation(block: false, alt: "2 n")[$2 n$] to both sides to leave only constants on the right.], [#figure(figph[A mathematical equation is displayed on a white background: '8n + 2n - 4 = -2n + 2n + 6'. The '+2n' terms are highlighted in red on both sides of the equation.], alt: "A mathematical equation is displayed on a white background: '8n + 2n - 4 = -2n + 2n + 6'. The '+2n' terms are highlighted in red on both sides of the equation.", caption: none)], [],
  [Combine like terms.], [#figure(figph[The image shows the mathematical equation '10n - 4 = 6' centered on a white background.], alt: "The image shows the mathematical equation '10n - 4 = 6' centered on a white background.", caption: none)], [],
  [We don’t want any constants on the left side, so add #math.equation(block: false, alt: "4")[$4$] to both sides.], [#figure(figph[A mathematical equation, '10n - 4 + 4 = 6 + 4', demonstrating the addition property of equality where '+4' is added to both sides of the equation, highlighted in red.], alt: "A mathematical equation, '10n - 4 + 4 = 6 + 4', demonstrating the addition property of equality where '+4' is added to both sides of the equation, highlighted in red.", caption: none)], [],
  [Simplify.], [#figure(figph[The equation 10n = 10 is displayed, where solving for 'n' would yield 1.], alt: "The equation 10n = 10 is displayed, where solving for 'n' would yield 1.", caption: none)], [],
  [The variable term is on the left and the constant term is on the right. To get the coefficient of #math.equation(block: false, alt: "n")[$n$] to be one, divide both sides by 10.], [#figure(figph[A mathematical equation showing 10n divided by 10 equals 10 divided by 10, with the denominator '10' highlighted in red on both sides.], alt: "A mathematical equation showing 10n divided by 10 equals 10 divided by 10, with the denominator '10' highlighted in red on both sides.", caption: none)], [],
  [Simplify.], [#figure(figph[The image features the mathematical expression 'n = 1' written in black characters against a plain white background, centrally placed in the frame.], alt: "The image features the mathematical expression 'n = 1' written in black characters against a plain white background, centrally placed in the frame.", caption: none)], [],
  [Check:], [#figure(figph[A mathematical equation is displayed: 8n - 4 = -2n + 6. It's an algebraic equation with one variable, 'n', on both sides of the equality.], alt: "A mathematical equation is displayed: 8n - 4 = -2n + 6. It's an algebraic equation with one variable, 'n', on both sides of the equality.", caption: none)], [],
  [Let #math.equation(block: false, alt: "n equals 1")[$n = 1$].], [#figure(figph[A mathematical equation is displayed: 8 \* 1 - 4 =? -2 \* 1 + 6. Both sides simplify to 4, making the statement true. The numbers '1' are highlighted in red.], alt: "A mathematical equation is displayed: 8 * 1 - 4 =? -2 * 1 + 6. Both sides simplify to 4, making the statement true. The numbers '1' are highlighted in red.", caption: none)], [],
  [], [#figure(figph[A mathematical equation 8 - 4 = -2 + 6 is displayed on a white background, with a question mark positioned above the equals sign.], alt: "A mathematical equation 8 - 4 = -2 + 6 is displayed on a white background, with a question mark positioned above the equals sign.", caption: none)], [],
  [], [#figure(figph[The equation '4=4' is displayed on a white background, followed by a checkmark, confirming its accuracy.], alt: "The equation '4=4' is displayed on a white background, followed by a checkmark, confirming its accuracy.", 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[
#math.equation(block: true, alt: "q equals 1")[$q = 1$]
]
]

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

#examplebox("Example 8")[][
Solve: #math.equation(block: false, alt: "7 a minus 3 equals 13 a plus 7 .")[$7 a − 3 = 13 a + 7 .$]

#solutionbox[
In the first step, choose the variable side by comparing the coefficients of the variables on each side.

Since #math.equation(block: false, alt: "13 greater than 7")[$13 > 7$], make the right side the “variable” side and the left side the “constant” side.

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[Algebraic equation 7a - 3 = 13a + 7. 'Constant' is labeled above -3 and +7, while 'variable' is above 7a and 13a, using red text to differentiate these fundamental algebraic terms.], alt: "Algebraic equation 7a - 3 = 13a + 7. 'Constant' is labeled above -3 and +7, while 'variable' is above 7a and 13a, using red text to differentiate these fundamental algebraic terms.", caption: none)], [],
  [Subtract #math.equation(block: false, alt: "7 a")[$7 a$] from both sides to remove the variable term from the left.], [#figure(figph[A mathematical equation is displayed: 7a - 7a - 3 = 13a - 7a + 7. Terms with '7a' are highlighted in red, indicating cancellation or a specific step in solving the equation.], alt: "A mathematical equation is displayed: 7a - 7a - 3 = 13a - 7a + 7. Terms with '7a' are highlighted in red, indicating cancellation or a specific step in solving the equation.", caption: none)], [],
  [Combine like terms.], [#figure(figph[A mathematical equation is displayed, stating '-3 = 6a + 7' in a white font against a black background. The equation involves a negative integer, a variable 'a' multiplied by a coefficient, and a positive integer, setting up a linear equation to be solved for 'a'.], alt: "A mathematical equation is displayed, stating '-3 = 6a + 7' in a white font against a black background. The equation involves a negative integer, a variable 'a' multiplied by a coefficient, and a positive integer, setting up a linear equation to be solved for 'a'.", caption: none)], [],
  [Subtract #math.equation(block: false, alt: "7")[$7$] from both sides to remove the constant from the right.], [#figure(figph[A mathematical equation shows '-3 - 7 = 6a + 7 - 7'. The numbers 7 on both sides of the equation are highlighted in red.], alt: "A mathematical equation shows '-3 - 7 = 6a + 7 - 7'. The numbers 7 on both sides of the equation are highlighted in red.", caption: none)], [],
  [Simplify.], [#figure(figph[The image displays a mathematical equation, -10 = 6a, in a clear and centered format against a plain white background, showing a simple algebraic problem.], alt: "The image displays a mathematical equation, -10 = 6a, in a clear and centered format against a plain white background, showing a simple algebraic problem.", caption: none)], [],
  [Divide both sides by #math.equation(block: false, alt: "6")[$6$] to make #math.equation(block: false, alt: "1")[$1$] the coefficient of #math.equation(block: false, alt: "a")[$a$].], [#figure(figph[A mathematical equation shows a fraction -10/6 equal to the fraction 6a/6. The denominators are both 6, colored in red, while the numerators -10 and 6a are in black.], alt: "A mathematical equation shows a fraction -10/6 equal to the fraction 6a/6. The denominators are both 6, colored in red, while the numerators -10 and 6a are in black.", caption: none)], [],
  [Simplify.], [#figure(figph[A mathematical equation displays '-5/3 = a' against a white background.], alt: "A mathematical equation displays '-5/3 = a' against a white background.", caption: none)], [],
  [Check:], [#figure(figph[An algebraic equation showing 7a minus 3 equals 13a plus 7.], alt: "An algebraic equation showing 7a minus 3 equals 13a plus 7.", caption: none)], [],
  [Let #math.equation(block: false, alt: "a equals minus the fraction 5 over 3")[$a = − frac(5, 3)$].], [#figure(figph[A mathematical equation is displayed on a white background, asking to verify the equality of 7(-5/3) - 3 and 13(-5/3) + 7. Both sides simplify to -44/3, confirming the equality.], alt: "A mathematical equation is displayed on a white background, asking to verify the equality of 7(-5/3) - 3 and 13(-5/3) + 7. Both sides simplify to -44/3, confirming the equality.", caption: none)], [],
  [], [#figure(figph[A mathematical equation is displayed: -35/3 - 9/3 = ? - 65/3 + 21/3. The '?' indicates a missing term, and the fractions all share a common denominator of 3, involving subtraction and addition operations.], alt: "A mathematical equation is displayed: -35/3 - 9/3 = ? - 65/3 + 21/3. The '?' indicates a missing term, and the fractions all share a common denominator of 3, involving subtraction and addition operations.", caption: none)], [],
  [], [#figure(figph[The equation -44/3 = -44/3 is shown, verified by a checkmark.], alt: "The equation -44/3 = -44/3 is shown, verified by a checkmark.", caption: none)], [],
))
]
]

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

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

In the last example, we could have made the left side the “variable” side, but it would have led to a negative coefficient on the variable term. (Try it!) While we could work with the negative, there is less chance of errors when working with positives. The strategy outlined above helps avoid the negatives!

To solve an equation with fractions, we just follow the steps of our strategy to get the solution!

#examplebox("Example 9")[][
Solve: #math.equation(block: false, alt: "the fraction 5 over 4 x plus 6 equals the fraction 1 over 4 x minus 2 .")[$frac(5, 4) x + 6 = frac(1, 4) x − 2 .$]

#solutionbox[
Since #math.equation(block: false, alt: "the fraction 5 over 4 greater than the fraction 1 over 4")[$frac(5, 4) > frac(1, 4)$], 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 showing variables (5/4x, 1/4x) and constants (6, -2) for solving.], alt: "An algebraic equation showing variables (5/4x, 1/4x) and constants (6, -2) for solving.", caption: none)]),
  [Subtract #math.equation(block: false, alt: "the fraction 1 over 4 x")[$frac(1, 4) x$] from both sides.], [#figure(figph[A mathematical equation is displayed: 5/4x - 1/4x + 6 = 1/4x - 1/4x - 2. The terms 1/4x are highlighted in red on both sides of the equation.], alt: "A mathematical equation is displayed: 5/4x - 1/4x + 6 = 1/4x - 1/4x - 2. The terms 1/4x are highlighted in red on both sides of the equation.", caption: none)],
  [Combine like terms.], [#figure(figph[A simple algebraic equation is displayed, reading 'x + 6 = -2' in white text against a white background.], alt: "A simple algebraic equation is displayed, reading 'x + 6 = -2' in white text against a white background.", caption: none)],
  [Subtract #math.equation(block: false, alt: "6")[$6$] from both sides.], [#figure(figph[A mathematical equation shows a step in solving for 'x', where 6 is subtracted from both sides of the equation x + 6 = -2, resulting in x + 6 - 6 = -2 - 6. The subtracted 6s are red.], alt: "A mathematical equation shows a step in solving for 'x', where 6 is subtracted from both sides of the equation x + 6 = -2, resulting in x + 6 - 6 = -2 - 6. The subtracted 6s are red.", caption: none)],
  [Simplify.], [#figure(figph[The image displays a simple mathematical equation, X = -8, written in a clear, sans-serif font against a plain white background.], alt: "The image displays a simple mathematical equation, X = -8, written in a clear, sans-serif font against a plain white background.", caption: none)],
  [#math.equation(block: false, alt: "Check:, the fraction 5 over 4 x plus 6, equals, the fraction 1 over 4 x minus 2; Let x equals −8 ., the fraction 5 over 4 open parenthesis −8 close parenthesis plus 6, equals ?, the fraction 1 over 4 open parenthesis −8 close parenthesis minus 2; −10 plus 6, equals ?, −2 minus 2; −4, equals, −4 ✓")[$"Check:" &  &  & frac(5, 4) x + 6 & = & frac(1, 4) x − 2 \ "Let" #h(0.2em) x = −8 . &  &  & frac(5, 4) ( −8 ) + 6 & limits(=)^(?) & frac(1, 4) ( −8 ) − 2 \  &  &  & −10 + 6 & limits(=)^(?) & −2 − 2 \  &  &  & −4 & = & −4 ✓$]], [],
))
]
]

#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 .")[$frac(7, 8) x − 12 = − frac(1, 8) x − 2 .$]

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

#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 .")[$frac(7, 6) y + 11 = frac(1, 6) y + 8 .$]

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

We will use the same strategy to find the solution for an equation with decimals.

#examplebox("Example 10")[][
Solve: #math.equation(block: false, alt: "7.8 x plus 4 equals 5.4 x minus 8 .")[$7.8 x + 4 = 5.4 x − 8 .$]

#solutionbox[
Since #math.equation(block: false, alt: "7.8 greater than 5.4")[$7.8 > 5.4$], make the left side the “variable” side and the right side the “constant” side.

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[An algebraic equation, 7.8x + 4 = 5.4x - 8, is shown with the left side designated as the 'variable side' and the right side as the 'constant side'.], alt: "An algebraic equation, 7.8x + 4 = 5.4x - 8, is shown with the left side designated as the 'variable side' and the right side as the 'constant side'.", caption: none)], [],
  [Subtract #math.equation(block: false, alt: "5.4 x")[$5.4 x$] from both sides.], [#figure(figph[A mathematical equation is displayed: 7.8x - 5.4x + 4 = 5.4x - 5.4x - 8. The term '5.4x' is highlighted in red, indicating a common or significant part of the expression.], alt: "A mathematical equation is displayed: 7.8x - 5.4x + 4 = 5.4x - 5.4x - 8. The term '5.4x' is highlighted in red, indicating a common or significant part of the expression.", caption: none)], [],
  [Combine like terms.], [#figure(figph[A mathematical equation on a white background reads '2.4x + 4 = -8'. The numbers and symbols are in a dark gray font.], alt: "A mathematical equation on a white background reads '2.4x + 4 = -8'. The numbers and symbols are in a dark gray font.", caption: none)], [],
  [Subtract #math.equation(block: false, alt: "4")[$4$] from both sides.], [#figure(figph[A mathematical equation shows 2.4x + 4 - 4 = -8 - 4, with the number 4 subtracted from both sides highlighted in red.], alt: "A mathematical equation shows 2.4x + 4 - 4 = -8 - 4, with the number 4 subtracted from both sides highlighted in red.", caption: none)], [],
  [Simplify.], [#figure(figph[A mathematical equation is displayed, showing '2.4x = -12' in a grey font against a plain white background.], alt: "A mathematical equation is displayed, showing '2.4x = -12' in a grey font against a plain white background.", caption: none)], [],
  [Use the Division Propery of Equality.], [#figure(figph[A mathematical equation illustrating the division of both sides by 2.4 to solve for x. The equation is (2.4x)/2.4 = -12/2.4, with the denominator 2.4 highlighted in red.], alt: "A mathematical equation illustrating the division of both sides by 2.4 to solve for x. The equation is (2.4x)/2.4 = -12/2.4, with the denominator 2.4 highlighted in red.", caption: none)], [],
  [Simplify.], [#figure(figph[The image displays the mathematical equation X = -5 centered on a plain white background, rendered in a grayscale font.], alt: "The image displays the mathematical equation X = -5 centered on a plain white background, rendered in a grayscale font.", caption: none)], [],
  [Check:], [#figure(figph[A mathematical equation is displayed, reading '7.8x + 4 = 5.4x - 5' against a white background.], alt: "A mathematical equation is displayed, reading '7.8x + 4 = 5.4x - 5' against a white background.", caption: none)], [],
  [Let #math.equation(block: false, alt: "x equals −5")[$x = −5$].], [#figure(figph[A mathematical equation is displayed: 7.8 multiplied by -5, plus 4, equals 5.4 multiplied by -5, minus 8. The numbers -5 are highlighted in red.], alt: "A mathematical equation is displayed: 7.8 multiplied by -5, plus 4, equals 5.4 multiplied by -5, minus 8. The numbers -5 are highlighted in red.", caption: none)], [],
  [], [#figure(figph[A mathematical problem displays the equation -39 + 4 = -27 - 8. A question mark hovers above the equal sign, challenging whether the equation holds true. Evaluating both sides confirms the equality, as both result in -35.], alt: "A mathematical problem displays the equation -39 + 4 = -27 - 8. A question mark hovers above the equal sign, challenging whether the equation holds true. Evaluating both sides confirms the equality, as both result in -35.", caption: none)], [],
  [], [#figure(figph[A mathematical equation showing '-35 = -35' followed by a checkmark, indicating that the equality is correct.], alt: "A mathematical equation showing '-35 = -35' followed by a checkmark, indicating that the equality is correct.", 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[
#math.equation(block: true, alt: "x equals −5")[$x = −5$]
]
]

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


=== Key Concepts

- #strong[Beginning Strategy for Solving an Equation with Variables and Constants on Both Sides of the Equation]
+ Choose which side will be the “variable” side—the other side will be the “constant” side.
+ Collect the variable terms to the “variable” side of the equation, using the Addition or Subtraction Property of Equality.
+ Collect all the constants to the other side of the equation, using the Addition or Subtraction Property of Equality.
+ Make the coefficient of the variable equal 1, using the Multiplication or Division Property of Equality.
+ Check the solution by substituting it into the original equation.


==== Practice Makes Perfect

#strong[Solve Equations with Constants on Both Sides]

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

#math.equation(block: true, alt: "9 x minus 3 equals 60")[$9 x − 3 = 60$]

#math.equation(block: true, alt: "12 x minus 8 equals 64")[$12 x − 8 = 64$]

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

#math.equation(block: true, alt: "14 w plus 5 equals 117")[$14 w + 5 = 117$]

#math.equation(block: true, alt: "15 y plus 7 equals 97")[$15 y + 7 = 97$]

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

#math.equation(block: true, alt: "2 a plus 8 equals −28")[$2 a + 8 = −28$]

#math.equation(block: true, alt: "3 m plus 9 equals −15")[$3 m + 9 = −15$]

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

#math.equation(block: true, alt: "−62 equals 8 n minus 6")[$−62 = 8 n − 6$]

#math.equation(block: true, alt: "−77 equals 9 b minus 5")[$−77 = 9 b − 5$]

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

#math.equation(block: true, alt: "35 equals −13 y plus 9")[$35 = −13 y + 9$]

#math.equation(block: true, alt: "60 equals −21 x minus 24")[$60 = −21 x − 24$]

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

#math.equation(block: true, alt: "−12 p minus 9 equals 9")[$−12 p − 9 = 9$]

#math.equation(block: true, alt: "−14 q minus 2 equals 16")[$−14 q − 2 = 16$]

#solutionbox[
#math.equation(block: true, alt: "q equals minus the fraction 9 over 7")[$q = − frac(9, 7)$]
]

#strong[Solve Equations with Variables on Both Sides]

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

#math.equation(block: true, alt: "19 z equals 18 z minus 7")[$19 z = 18 z − 7$]

#math.equation(block: true, alt: "21 k equals 20 k minus 11")[$21 k = 20 k − 11$]

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

#math.equation(block: true, alt: "9 x plus 36 equals 15 x")[$9 x + 36 = 15 x$]

#math.equation(block: true, alt: "8 x plus 27 equals 11 x")[$8 x + 27 = 11 x$]

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

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

#math.equation(block: true, alt: "9 q equals 44 minus 2 q")[$9 q = 44 − 2 q$]

#math.equation(block: true, alt: "5 z equals 39 minus 8 z")[$5 z = 39 − 8 z$]

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

#math.equation(block: true, alt: "6 y plus the fraction 1 over 2 equals 5 y")[$6 y + frac(1, 2) = 5 y$]

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

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

#math.equation(block: true, alt: "−18 a minus 8 equals −22 a")[$−18 a − 8 = −22 a$]

#math.equation(block: true, alt: "−11 r minus 8 equals −7 r")[$−11 r − 8 = −7 r$]

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

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

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

#math.equation(block: true, alt: "8 x minus 15 equals 7 x plus 3")[$8 x − 15 = 7 x + 3$]

#math.equation(block: true, alt: "6 x minus 17 equals 5 x plus 2")[$6 x − 17 = 5 x + 2$]

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

#math.equation(block: true, alt: "26 plus 13 d equals 14 d plus 11")[$26 + 13 d = 14 d + 11$]

#math.equation(block: true, alt: "21 plus 18 f equals 19 f plus 14")[$21 + 18 f = 19 f + 14$]

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

#math.equation(block: true, alt: "2 p minus 1 equals 4 p minus 33")[$2 p − 1 = 4 p − 33$]

#math.equation(block: true, alt: "12 q minus 5 equals 9 q minus 20")[$12 q − 5 = 9 q − 20$]

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

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

#math.equation(block: true, alt: "8 c plus 7 equals −3 c minus 37")[$8 c + 7 = −3 c − 37$]

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

#math.equation(block: true, alt: "5 y minus 30 equals −5 y plus 30")[$5 y − 30 = −5 y + 30$]

#math.equation(block: true, alt: "7 x minus 17 equals −8 x plus 13")[$7 x − 17 = −8 x + 13$]

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

#math.equation(block: true, alt: "7 s plus 12 equals 5 plus 4 s")[$7 s + 12 = 5 + 4 s$]

#math.equation(block: true, alt: "9 p plus 14 equals 6 plus 4 p")[$9 p + 14 = 6 + 4 p$]

#solutionbox[
#math.equation(block: true, alt: "p equals minus the fraction 8 over 5")[$p = − frac(8, 5)$]
]

#math.equation(block: true, alt: "2 z minus 6 equals 23 minus z")[$2 z − 6 = 23 − z$]

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

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

#math.equation(block: true, alt: "the fraction 5 over 3 c minus 3 equals the fraction 2 over 3 c minus 16")[$frac(5, 3) c − 3 = frac(2, 3) c − 16$]

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

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

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

#math.equation(block: true, alt: "11 minus the fraction 1 over 5 a equals the fraction 4 over 5 a plus 4")[$11 − frac(1, 5) a = frac(4, 5) a + 4$]

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

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

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

#solutionbox[
#math.equation(block: true, alt: "a equals −40")[$a = −40$]
]

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

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

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

#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[
#math.equation(block: true, alt: "z equals 3.46")[$z = 3.46$]
]

#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[
#math.equation(block: true, alt: "w equals 60")[$w = 60$]
]

#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[
#math.equation(block: true, alt: "x equals 23")[$x = 23$]
]


==== Everyday Math

#strong[Concert tickets] At a school concert the total value of tickets sold was \$1506. Student tickets sold for \$6 and adult tickets sold for \$9. The number of adult tickets sold was 5 less than 3 times the number of student tickets. Find the number of student tickets sold, #emph[s], by solving the equation #math.equation(block: false, alt: "6 s plus 27 s minus 45 equals 1506")[$6 s + 27 s − 45 = 1506$].

#strong[Making a fence] Jovani has 150 feet of fencing to make a rectangular garden in his backyard. He wants the length to be 15 feet more than the width. Find the width, #emph[w], by solving the equation #math.equation(block: false, alt: "150 equals 2 w plus 30 plus 2 w")[$150 = 2 w + 30 + 2 w$].

#solutionbox[
30 feet
]


==== Writing Exercises

Solve the equation #math.equation(block: false, alt: "the fraction 6 over 5 y minus 8 equals the fraction 1 over 5 y plus 7")[$frac(6, 5) y − 8 = frac(1, 5) y + 7$] explaining all the steps of your solution as in the examples in this section.

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 as in the examples in this section.

#solutionbox[
#math.equation(block: false, alt: "x equals 2")[$x = 2$] Justifications will vary.
]

When solving an equation with variables on both sides, why is it usually better to choose the side with the larger coefficient of #math.equation(block: false, alt: "x")[$x$] to be the “variable” side?

Is #math.equation(block: false, alt: "x equals −2")[$x = −2$] a solution to the equation #math.equation(block: false, alt: "5 minus 2 x equals −4 x plus 1")[$5 − 2 x = −4 x + 1$]
? How do you know?

#solutionbox[
Yes. Justifications will vary.
]


==== Self Check

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

#figure(figph[This is a table that has four rows and four columns. In the first row, which is a header row, the cells read from left to right: “I can...,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can...” reads: “solve an equation with constants on both sides,” “solve an equation with variables on both sides,” and “solve an equation with variables and constants on both sides. ” The rest of the cells are blank.], alt: "This is a table that has four rows and four columns. In the first row, which is a header row, the cells read from left to right: “I can...,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can...” reads: “solve an equation with constants on both sides,” “solve an equation with variables on both sides,” and “solve an equation with variables and constants on both sides. ” The rest of the cells are blank.", caption: none)

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