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== 8.2#h(0.6em)Solve Equations Using the Division and Multiplication Properties of Equality

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

Simplify: #math.equation(block: false, alt: "−7 open parenthesis the fraction 1 over −7 close parenthesis .")[$−7 ( frac(1, −7) ) .$] #linebreak() 
If you missed this problem, review .

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
What is the reciprocal of #math.equation(block: false, alt: "minus the fraction 3 over 8 ?")[$− frac(3, 8) ?$] #linebreak() 
If you missed this problem, review .

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Evaluate #math.equation(block: false, alt: "9 x plus 2")[$9 x + 2$] when #math.equation(block: false, alt: "x equals −3 .")[$x = −3 .$] #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "minus 25")[$− 25$]
]
]


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

We introduced the Multiplication and Division Properties of Equality in Solve Equations Using Integers; The Division Property of Equality and Solve Equations with Fractions. We modeled how these properties worked using envelopes and counters and then applied them to solving equations (See Solve Equations Using Integers; The Division Property of Equality). We restate them again here as we prepare to use these properties again.

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

#strong[Division Property of Equality]: For all real numbers #math.equation(block: false, alt: "a , b , c ,")[$a , b , c ,$] and #math.equation(block: false, alt: "c not equal to 0 ,")[$c ≠ 0 ,$] if #math.equation(block: false, alt: "a equals b ,")[$a = b ,$] then #math.equation(block: false, alt: "the fraction a over c equals the fraction b over c .")[$frac(a, c) = frac(b, c) .$]

#strong[Multiplication Property of Equality]: For all real numbers #math.equation(block: false, alt: "a , b , c ,")[$a , b , c ,$] if #math.equation(block: false, alt: "a equals b ,")[$a = b ,$] then #math.equation(block: false, alt: "a c equals b c .")[$a c = b c .$]
]

When you divide or multiply both sides of an equation by the same quantity, you still have equality.

Let’s review how these properties of equality can be applied in order to solve equations. Remember, the goal is to ‘undo’ the operation on the variable. In the example below the variable is multiplied by #math.equation(block: false, alt: "4 ,")[$4 ,$] so we will divide both sides by #math.equation(block: false, alt: "4")[$4$] to ‘undo’ the multiplication.

#examplebox("Example 1")[][
Solve: #math.equation(block: false, alt: "4 x equals −28 .")[$4 x = −28 .$]

#solutionbox[
We use the Division Property of Equality to divide both sides by #math.equation(block: false, alt: "4 .")[$4 .$]

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation displays '4x = -28' in black font against a white background.], alt: "A mathematical equation displays '4x = -28' in black font against a white background.", caption: none)]),
  [Divide both sides by 4 to undo the multiplication.], [#figure(figph[A mathematical equation showing 4x divided by 4 equals -28 divided by 4. The number 4 in the denominators is highlighted in red, indicating division on both sides of the equation.], alt: "A mathematical equation showing 4x divided by 4 equals -28 divided by 4. The number 4 in the denominators is highlighted in red, indicating division on both sides of the equation.", caption: none)],
  [Simplify.], [#figure(figph[The equation x = -7 is displayed in black text on a plain white background, presenting a simple algebraic expression.], alt: "The equation x = -7 is displayed in black text on a plain white background, presenting a simple algebraic expression.", caption: none)],
  [Check your answer. Let #math.equation(block: false, alt: "x equals −7")[$x = −7$].], [],
  [#figure(figph[The image shows the mathematical equation 4x = -28, rendered in a bold, metallic-looking font against a white background.], alt: "The image shows the mathematical equation 4x = -28, rendered in a bold, metallic-looking font against a white background.", caption: none)], [],
  [#figure(figph[A mathematical equation, 4 multiplied by -7, is shown with a question mark above the equals sign, asking to verify if the product is -28. The statement 4(-7) = -28 is true.], alt: "A mathematical equation, 4 multiplied by -7, is shown with a question mark above the equals sign, asking to verify if the product is -28. The statement 4(-7) = -28 is true.", caption: none)], [],
  [#figure(figph[A mathematical equation displays '-28 = -28' followed by a checkmark, indicating that the statement is true and verified.], alt: "A mathematical equation displays '-28 = -28' followed by a checkmark, indicating that the statement is true and verified.", caption: none)], [],
))

Since this is a true statement, #math.equation(block: false, alt: "x equals −7")[$x = −7$] is a solution to #math.equation(block: false, alt: "4 x equals −28 .")[$4 x = −28 .$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "3 y equals −48 .")[$3 y = −48 .$]

#solutionbox[
#emph[y] = −16
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "4 z equals −52 .")[$4 z = −52 .$]

#solutionbox[
#emph[z] = −13
]
]

In the previous example, to ‘undo’ multiplication, we divided. How do you think we ‘undo’ division?

#examplebox("Example 2")[][
Solve: #math.equation(block: false, alt: "the fraction a over −7 equals −42 .")[$frac(#h(0.4em) a, −7) = −42 .$]

#solutionbox[
Here #math.equation(block: false, alt: "a")[$a$] is divided by #math.equation(block: false, alt: "−7 .")[$−7 .$] We can multiply both sides by #math.equation(block: false, alt: "−7")[$−7$] to isolate #math.equation(block: false, alt: "a .")[$a .$]

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation showing 'a' divided by '-7' equals '-42'. The equation is written as a fraction: a over -7 = -42.], alt: "A mathematical equation showing 'a' divided by '-7' equals '-42'. The equation is written as a fraction: a over -7 = -42.", caption: none)]),
  [Multiply both sides by #math.equation(block: false, alt: "−7")[$−7$].], [#figure(figph[A mathematical equation shows '-7 multiplied by (a divided by -7) equals -7 multiplied by -42'. The -7 on both sides of the equation is highlighted in red.], alt: "A mathematical equation shows '-7 multiplied by (a divided by -7) equals -7 multiplied by -42'. The -7 on both sides of the equation is highlighted in red.", caption: none)

 #linebreak() 
    #figure(figph[A mathematical equation is displayed, showing -7a divided by -7 equals 294.], alt: "A mathematical equation is displayed, showing -7a divided by -7 equals 294.", caption: none)],
  [Simplify.], [#figure(figph[The variable 'a' is assigned the value 294.], alt: "The variable 'a' is assigned the value 294.", caption: none)],
  [Check your answer. Let #math.equation(block: false, alt: "a equals 294")[$a = 294$].], [],
  [#figure(figph[A mathematical equation is displayed on a white background. The equation reads as 'a divided by negative seven equals negative forty-two' (a/-7 = -42).], alt: "A mathematical equation is displayed on a white background. The equation reads as 'a divided by negative seven equals negative forty-two' (a/-7 = -42).", caption: none)], [],
  [#figure(figph[A mathematical equation shows a fraction 294 divided by -7, with a question mark over the equals sign, comparing the result to -42. This asks whether 294 / -7 is indeed equal to -42.], alt: "A mathematical equation shows a fraction 294 divided by -7, with a question mark over the equals sign, comparing the result to -42. This asks whether 294 / -7 is indeed equal to -42.", caption: none)], [],
  [#figure(figph[A mathematical equation displays '-42 = -42' with a checkmark next to it, indicating that the statement is correct and verified.], alt: "A mathematical equation displays '-42 = -42' with a checkmark next to it, indicating that the statement is correct and verified.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the fraction b over −6 equals −24 .")[$frac(#h(0.4em) b, −6) = −24 .$]

#solutionbox[
#emph[b] = 144
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the fraction c over −8 equals −16 .")[$frac(#h(0.4em) c, −8) = −16 .$]

#solutionbox[
#emph[c] = 128
]
]

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

#solutionbox[
Remember #math.equation(block: false, alt: "minus r")[$− r$] is equivalent to #math.equation(block: false, alt: "−1 r .")[$−1 r .$]

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[A mathematical equation is displayed with a black text on a white background. The equation reads '-r = 2'.], alt: "A mathematical equation is displayed with a black text on a white background. The equation reads '-r = 2'.", caption: none)], [],
  [Rewrite #math.equation(block: false, alt: "minus r")[$− r$] as #math.equation(block: false, alt: "−1 r")[$−1 r$].], [#figure(figph[A mathematical equation is displayed on a white background, reading '-1r = 2' in bold, black characters, appearing as if a variable 'r' is being solved for.], alt: "A mathematical equation is displayed on a white background, reading '-1r = 2' in bold, black characters, appearing as if a variable 'r' is being solved for.", caption: none)], [],
  [Divide both sides by #math.equation(block: false, alt: "−1")[$−1$].], [#figure(figph[The equation shows dividing -1r by -1 on the left side and 2 by -1 on the right side, resulting in r = -2.], alt: "The equation shows dividing -1r by -1 on the left side and 2 by -1 on the right side, resulting in r = -2.", caption: none)], [],
  [], [#figure(figph[The image shows the mathematical equation 'r = -2' displayed in a bold, black font against a white background.], alt: "The image shows the mathematical equation 'r = -2' displayed in a bold, black font against a white background.", caption: none)], [],
  [Check.], [#figure(figph[The image displays a mathematical equation written in black text on a white background: -r=2.], alt: "The image displays a mathematical equation written in black text on a white background: -r=2.", caption: none)], [],
  [Substitute #math.equation(block: false, alt: "r equals −2")[$r = −2$]], [#figure(figph[Mathematical expression -(-2) =? 2, questioning if a double negative results in a positive. It does, so the statement is true.], alt: "Mathematical expression -(-2) =? 2, questioning if a double negative results in a positive. It does, so the statement is true.", caption: none)], [],
  [Simplify.], [#figure(figph[A simple equation '2=2' with a checkmark, denoting a verified truth.], alt: "A simple equation '2=2' with a checkmark, denoting a verified truth.", caption: none)], [],
))

In Solve Equations with Fractions, we saw that there are two other ways to solve #math.equation(block: false, alt: "minus r equals 2 .")[$− r = 2 .$]

We could multiply both sides by #math.equation(block: false, alt: "−1 .")[$−1 .$]

We could take the opposite of both sides.
]
]

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

#solutionbox[
#emph[k] = −8
]
]

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

#solutionbox[
#emph[g] = −3
]
]

#examplebox("Example 4")[][
Solve: #math.equation(block: false, alt: "the fraction 2 over 3 x equals 18 .")[$frac(2, 3) #h(0.1em) x = 18 .$]

#solutionbox[
Since the product of a number and its reciprocal is #math.equation(block: false, alt: "1 ,")[$1 ,$] our strategy will be to isolate #math.equation(block: false, alt: "x")[$x$] by multiplying by the reciprocal of #math.equation(block: false, alt: "the fraction 2 over 3 .")[$frac(2, 3) .$]

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed on a white background, reading '(2/3)x = 18'.], alt: "A mathematical equation is displayed on a white background, reading '(2/3)x = 18'.", caption: none)]),
  [Multiply by the reciprocal of #math.equation(block: false, alt: "the fraction 2 over 3")[$frac(2, 3)$].], [#figure(figph[An algebraic equation showing 3/2 multiplied by 2/3x equals 3/2 multiplied by 18, demonstrating the isolation of 'x' by multiplying both sides by the reciprocal 3/2, highlighted in red.], alt: "An algebraic equation showing 3/2 multiplied by 2/3x equals 3/2 multiplied by 18, demonstrating the isolation of 'x' by multiplying both sides by the reciprocal 3/2, highlighted in red.", caption: none)],
  [Reciprocals multiply to one.], [#figure(figph[A mathematical equation shows '1x = 3/2 multiplied by 18/1' on a white background.], alt: "A mathematical equation shows '1x = 3/2 multiplied by 18/1' on a white background.", caption: none)],
  [Multiply.], [#figure(figph[The equation 'x = 27' is displayed in a black, sans-serif font on a plain white background, centrally positioned within the frame.], alt: "The equation 'x = 27' is displayed in a black, sans-serif font on a plain white background, centrally positioned within the frame.", caption: none)],
  [Check your answer. Let #math.equation(block: false, alt: "x equals 27")[$x = 27$]], [],
  [#figure(figph[A mathematical equation shows (2/3)x = 18.], alt: "A mathematical equation shows (2/3)x = 18.", caption: none)], [],
  [#figure(figph[A mathematical expression shows 2/3 multiplied by 27 (in red), followed by an equals sign with a question mark above it, and then the number 18, prompting a check of the equality.], alt: "A mathematical expression shows 2/3 multiplied by 27 (in red), followed by an equals sign with a question mark above it, and then the number 18, prompting a check of the equality.", caption: none)], [],
  [#figure(figph[The image displays the equation '18 = 18' followed by a checkmark, confirming its accuracy.], alt: "The image displays the equation '18 = 18' followed by a checkmark, confirming its accuracy.", caption: none)], [],
))

Notice that we could have divided both sides of the equation #math.equation(block: false, alt: "the fraction 2 over 3 x equals 18")[$frac(2, 3) #h(0.1em) x = 18$] by #math.equation(block: false, alt: "the fraction 2 over 3")[$frac(2, 3)$] to isolate #math.equation(block: false, alt: "x .")[$x .$] While this would work, multiplying by the reciprocal requires fewer steps.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the fraction 2 over 5 n equals 14 .")[$frac(2, 5) #h(0.1em) n = 14 .$]

#solutionbox[
#emph[n] = 35
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the fraction 5 over 6 y equals 15 .")[$frac(5, 6) #h(0.1em) y = 15 .$]

#solutionbox[
#emph[y] = 18
]
]


=== Solve Equations That Need to be Simplified

Many equations start out more complicated than the ones we’ve just solved. First, we need to simplify both sides of the equation as much as possible

#examplebox("Example 5")[][
Solve: #math.equation(block: false, alt: "8 x plus 9 x minus 5 x equals −3 plus 15 .")[$8 x + 9 x − 5 x = −3 + 15 .$]

#solutionbox[
Start by combining like terms to simplify each side.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed, reading '8x + 9x - 5x = -3 + 15'.], alt: "A mathematical equation is displayed, reading '8x + 9x - 5x = -3 + 15'.", caption: none)]),
  [Combine like terms.], [#figure(figph[The mathematical equation 12x = 12 is displayed in black text on a white background.], alt: "The mathematical equation 12x = 12 is displayed in black text on a white background.", caption: none)],
  [Divide both sides by 12 to isolate x.], [#figure(figph[An algebraic equation showing a step in solving for x: 12x/12 = 12/12, where both sides are divided by 12, with the denominator 12 highlighted in red.], alt: "An algebraic equation showing a step in solving for x: 12x/12 = 12/12, where both sides are divided by 12, with the denominator 12 highlighted in red.", caption: none)],
  [Simplify.], [#figure(figph[The equation 'x=1' is displayed in a serif font, rendered in black text against a plain white background.], alt: "The equation 'x=1' is displayed in a serif font, rendered in black text against a plain white background.", caption: none)],
  [Check your answer. Let #math.equation(block: false, alt: "x equals 1")[$x = 1$]], [],
  [#figure(figph[The image displays the algebraic equation 8x + 9x - 5x = -3 + 15.], alt: "The image displays the algebraic equation 8x + 9x - 5x = -3 + 15.", caption: none)], [],
  [#figure(figph[A mathematical equation, '8 ×× ×× 1 + 9 ×× ×× 1 - 5 ×× ×× 1 = -3 + 15', with a question mark above the equals sign, indicating a verification task.], alt: "A mathematical equation, '8 ×× ×× 1 + 9 ×× ×× 1 - 5 ×× ×× 1 = -3 + 15', with a question mark above the equals sign, indicating a verification task.", caption: none)], [],
  [#figure(figph[A math problem displays '8 + 9 - 5 = -3 + 15' with a question mark above the equals sign, asking if the statement is true. Calculating both sides reveals that 12 = 12, making the equation true.], alt: "A math problem displays '8 + 9 - 5 = -3 + 15' with a question mark above the equals sign, asking if the statement is true. Calculating both sides reveals that 12 = 12, making the equation true.", caption: none)], [],
  [#figure(figph[A mathematical equation '12 = 12' is displayed, followed by a checkmark, indicating correctness or validation.], alt: "A mathematical equation '12 = 12' is displayed, followed by a checkmark, indicating correctness or validation.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "7 x plus 6 x minus 4 x equals −8 plus 26 .")[$7 x + 6 x − 4 x = −8 + 26 .$]

#solutionbox[
#emph[x] = 2
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "11 n minus 3 n minus 6 n equals 7 minus 17 .")[$11 n − 3 n − 6 n = 7 − 17 .$]

#solutionbox[
#emph[n] = −5
]
]

#examplebox("Example 6")[][
Solve: #math.equation(block: false, alt: "11 minus 20 equals 17 y minus 8 y minus 6 y .")[$11 − 20 = 17 y − 8 y − 6 y .$]

#solutionbox[
Simplify each side by combining like terms.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed, showing '11 - 20 = 17y - 8y - 6y' in black text on a white background, demonstrating an algebraic problem to be solved for the variable 'y'.], alt: "A mathematical equation is displayed, showing '11 - 20 = 17y - 8y - 6y' in black text on a white background, demonstrating an algebraic problem to be solved for the variable 'y'.", caption: none)]),
  [Simplify each side.], [#figure(figph[A mathematical equation is displayed against a white background, which reads '-9 = 3y' in black text.], alt: "A mathematical equation is displayed against a white background, which reads '-9 = 3y' in black text.", caption: none)],
  [Divide both sides by 3 to isolate y.], [#figure(figph[The algebraic equation -9/3 = 3y/3, demonstrating division by 3 on both sides. The number 3 in the denominator is highlighted in red.], alt: "The algebraic equation -9/3 = 3y/3, demonstrating division by 3 on both sides. The number 3 in the denominator is highlighted in red.", caption: none)],
  [Simplify.], [#figure(figph[-3 = y is an equation showing that the variable 'y' is equal to the constant value -3. It represents a horizontal line in a Cartesian coordinate system, where all points on the line have a y-coordinate of -3.], alt: "-3 = y is an equation showing that the variable 'y' is equal to the constant value -3. It represents a horizontal line in a Cartesian coordinate system, where all points on the line have a y-coordinate of -3.", caption: none)],
  [Check your answer. Let #math.equation(block: false, alt: "y equals −3")[$y = −3$]], [],
  [#figure(figph[A mathematical equation is displayed, reading '11 - 20 = 17y - 8y - 6y' on a white background.], alt: "A mathematical equation is displayed, reading '11 - 20 = 17y - 8y - 6y' on a white background.", caption: none)], [],
  [#figure(figph[A mathematical equation asks whether 11 - 20 is equal to 17(-3) - 8(-3) - 6(-3), with a question mark positioned above the equality sign.], alt: "A mathematical equation asks whether 11 - 20 is equal to 17(-3) - 8(-3) - 6(-3), with a question mark positioned above the equality sign.", caption: none)], [],
  [#figure(figph[A mathematical equation with a question mark over the equals sign asks to verify if 11 - 20 is equal to -51 + 24 + 18.], alt: "A mathematical equation with a question mark over the equals sign asks to verify if 11 - 20 is equal to -51 + 24 + 18.", caption: none)], [],
  [#figure(figph[The equation -9 = -9 is shown with a checkmark, indicating its correctness.], alt: "The equation -9 = -9 is shown with a checkmark, indicating its correctness.", caption: none)], [],
))

Notice that the variable ended up on the right side of the equal sign when we solved the equation. You may prefer to take one more step to write the solution with the variable on the left side of the equal sign.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "18 minus 27 equals 15 c minus 9 c minus 3 c .")[$18 − 27 = 15 c − 9 c − 3 c .$]

#solutionbox[
#emph[c] = −3
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "18 minus 22 equals 12 x minus x minus 4 x .")[$18 − 22 = 12 x − x − 4 x .$]

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

#examplebox("Example 7")[][
Solve: #math.equation(block: false, alt: "−3 open parenthesis n minus 2 close parenthesis minus 6 equals 21 .")[$−3 ( n − 2 ) − 6 = 21 .$]

#solutionbox[
Remember—always simplify each side first.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed: -3(n - 2) - 6 = 21. This algebraic expression involves multiplication, subtraction, and a variable 'n', set equal to 21.], alt: "A mathematical equation is displayed: -3(n - 2) - 6 = 21. This algebraic expression involves multiplication, subtraction, and a variable 'n', set equal to 21.", caption: none)]),
  [Distribute.], [#figure(figph[A mathematical equation displays '-3n + 6 - 6 = 21' in black text on a white background.], alt: "A mathematical equation displays '-3n + 6 - 6 = 21' in black text on a white background.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation shows '-3n = 21' in black text against a white background.], alt: "A mathematical equation shows '-3n = 21' in black text against a white background.", caption: none)],
  [Divide both sides by -3 to isolate n.], [#figure(figph[A mathematical equation shows '-3n divided by -3 equals 21 divided by -3'. The denominator, -3, is highlighted in red on both sides of the equation, indicating division as a step in solving for 'n'.], alt: "A mathematical equation shows '-3n divided by -3 equals 21 divided by -3'. The denominator, -3, is highlighted in red on both sides of the equation, indicating division as a step in solving for 'n'.", caption: none)

 #linebreak() 
   #figure(figph[The mathematical equation 'n = -7' is displayed in bold black text on a plain white background.], alt: "The mathematical equation 'n = -7' is displayed in bold black text on a plain white background.", caption: none)],
  [Check your answer. Let #math.equation(block: false, alt: "n equals −7")[$n = −7$].], [],
  [#figure(figph[A mathematical equation is displayed, showing -3(n-2) - 6 = 21, centered against a white background.], alt: "A mathematical equation is displayed, showing -3(n-2) - 6 = 21, centered against a white background.", caption: none)], [],
  [#figure(figph[A mathematical equation showing -3 multiplied by the quantity (-7 minus 2), then minus 6, with a question mark over the equals sign before 21, asking if the expression equals 21.], alt: "A mathematical equation showing -3 multiplied by the quantity (-7 minus 2), then minus 6, with a question mark over the equals sign before 21, asking if the expression equals 21.", caption: none)], [],
  [#figure(figph[A mathematical equation is displayed: -3(-9) - 6 =? 21. It asks to verify if the expression on the left equals 21.], alt: "A mathematical equation is displayed: -3(-9) - 6 =? 21. It asks to verify if the expression on the left equals 21.", caption: none)], [],
  [#figure(figph[A mathematical equation '27 - 6 =? 21' is displayed, with a question mark positioned above the equals sign, implying an inquiry into the truth of the statement.], alt: "A mathematical equation '27 - 6 =? 21' is displayed, with a question mark positioned above the equals sign, implying an inquiry into the truth of the statement.", caption: none)], [],
  [#figure(figph[A mathematical equation '21 = 21' is displayed on a white background, followed by a black checkmark, confirming the equality's correctness.], alt: "A mathematical equation '21 = 21' is displayed on a white background, followed by a black checkmark, confirming the equality's correctness.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "−4 open parenthesis n minus 2 close parenthesis minus 8 equals 24 .")[$−4 ( n − 2 ) − 8 = 24 .$]

#solutionbox[
#emph[n] = −6
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "−6 open parenthesis n minus 2 close parenthesis minus 12 equals 30 .")[$−6 ( n − 2 ) − 12 = 30 .$]

#solutionbox[
#emph[n] = −5
]
]

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

- #link("https://www.openstax.org/l/24OneStepMultiL")[Solving One Step Equation by Mult/Div. Integers (Var on Left)]
- #link("https://www.openstax.org/l/24OneStepMultiR")[Solving One Step Equation by Mult/Div. Integers (Var on Right)]
- #link("https://www.openstax.org/l/24xa")[Solving One Step Equation in the Form: −#emph[x] = −#emph[a]]
]


=== Key Concepts

- #emph[Division and Multiplication Properties of Equality]- #emph[Division Property of Equality:] For all real numbers #emph[a, b, c,] and #math.equation(block: false, alt: "c not equal to 0")[$c ≠ 0$], if #math.equation(block: false, alt: "a equals b")[$a = b$], then #math.equation(block: false, alt: "the fraction a over c equals the fraction b over c")[$frac(a, c) = frac(b, c)$].
- #emph[Multiplication Property of Equality:] For all real numbers #emph[a, b, c,] if #math.equation(block: false, alt: "a equals b")[$a = b$], then #math.equation(block: false, alt: "a c equals b c")[$a c = b c$].


==== Practice Makes Perfect

#strong[Solve Equations Using the Division and Multiplication Properties of Equality]

In the following exercises, solve each equation for the variable using the Division Property of Equality and check the solution.

#math.equation(block: true, alt: "8 x equals 32")[$8 x = 32$]

#math.equation(block: true, alt: "7 p equals 63")[$7 p = 63$]

#solutionbox[
#emph[p] = 9
]

#math.equation(block: true, alt: "−5 c equals 55")[$−5 c = 55$]

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

#solutionbox[
#emph[x] = 3
]

#math.equation(block: true, alt: "−90 equals 6 y")[$−90 = 6 y$]

#math.equation(block: true, alt: "−72 equals 12 y")[$−72 = 12 y$]

#solutionbox[
#emph[y] = −6
]

#math.equation(block: true, alt: "−16 p equals −64")[$−16 p = −64$]

#math.equation(block: true, alt: "−8 m equals −56")[$−8 m = −56$]

#solutionbox[
#emph[m] = 7
]

#math.equation(block: true, alt: "0.25 z equals 3.25")[$0.25 z = 3.25$]

#math.equation(block: true, alt: "0.75 a equals 11.25")[$0.75 a = 11.25$]

#solutionbox[
#emph[a] = 15
]

#math.equation(block: true, alt: "−3 x equals 0")[$−3 x = 0$]

#math.equation(block: true, alt: "4 x equals 0")[$4 x = 0$]

#solutionbox[
#emph[x] = 0
]

In the following exercises, solve each equation for the variable using the Multiplication Property of Equality and check the solution.

#math.equation(block: true, alt: "the fraction x over 4 equals 15")[$frac(x, 4) = 15$]

#math.equation(block: true, alt: "the fraction z over 2 equals 14")[$frac(z, 2) = 14$]

#solutionbox[
#emph[z] = 28
]

#math.equation(block: true, alt: "−20 equals the fraction q over −5")[$−20 = frac(#h(0.4em) q, −5)$]

#math.equation(block: true, alt: "the fraction c over −3 equals −12")[$frac(#h(0.4em) c, −3) = −12$]

#solutionbox[
#emph[c] = 36
]

#math.equation(block: true, alt: "the fraction y over 9 equals −6")[$frac(y, 9) = −6$]

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

#solutionbox[
#emph[q] = −48
]

#math.equation(block: true, alt: "the fraction m over −12 equals 5")[$frac(m, −12) = 5$]

#math.equation(block: true, alt: "−4 equals the fraction p over −20")[$−4 = frac(p, −20)$]

#solutionbox[
#emph[p] = 80
]

#math.equation(block: true, alt: "the fraction 2 over 3 y equals 18")[$frac(2, 3) #h(0.1em) y = 18$]

#math.equation(block: true, alt: "the fraction 3 over 5 r equals 15")[$frac(3, 5) #h(0.1em) r = 15$]

#solutionbox[
#emph[r] = 25
]

#math.equation(block: true, alt: "minus the fraction 5 over 8 w equals 40")[$− frac(5, 8) #h(0.1em) w = 40$]

#math.equation(block: true, alt: "24 equals minus the fraction 3 over 4 x")[$24 = − frac(3, 4) #h(0.1em) x$]

#solutionbox[
#emph[x] = −32
]

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

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

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

#strong[Solve Equations That Need to be Simplified]

In the following exercises, solve the equation.

#math.equation(block: true, alt: "8 a plus 3 a minus 6 a equals −17 plus 27")[$8 a + 3 a − 6 a = −17 + 27$]

#math.equation(block: true, alt: "6 y minus 3 y plus 12 y equals −43 plus 28")[$6 y − 3 y + 12 y = −43 + 28$]

#solutionbox[
#emph[y] = −1
]

#math.equation(block: true, alt: "−9 x minus 9 x plus 2 x equals 50 minus 2")[$−9 x − 9 x + 2 x = 50 − 2$]

#math.equation(block: true, alt: "−5 m plus 7 m minus 8 m equals −6 plus 36")[$−5 m + 7 m − 8 m = −6 + 36$]

#solutionbox[
#emph[m] = −5
]

#math.equation(block: true, alt: "100 minus 16 equals 4 p minus 10 p minus p")[$100 − 16 = 4 p − 10 p − p$]

#math.equation(block: true, alt: "−18 minus 7 equals 5 t minus 9 t minus 6 t")[$−18 − 7 = 5 t − 9 t − 6 t$]

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

#math.equation(block: true, alt: "the fraction 7 over 8 n minus the fraction 3 over 4 n equals 9 plus 2")[$frac(7, 8) #h(0.1em) n − frac(3, 4) #h(0.1em) n = 9 + 2$]

#math.equation(block: true, alt: "the fraction 5 over 12 q plus the fraction 1 over 2 q equals 25 minus 3")[$frac(5, 12) #h(0.1em) q + frac(1, 2) #h(0.1em) q = 25 − 3$]

#solutionbox[
#emph[q] = 24
]

#math.equation(block: true, alt: "0.25 d plus 0.10 d equals 6 minus 0.75")[$0.25 d + 0.10 d = 6 − 0.75$]

#math.equation(block: true, alt: "0.05 p minus 0.01 p equals 2 plus 0.24")[$0.05 p − 0.01 p = 2 + 0.24$]

#solutionbox[
#emph[p] = 56
]


==== Everyday Math

#strong[Balloons] Ramona bought #math.equation(block: false, alt: "18")[$18$] balloons for a party. She wants to make #math.equation(block: false, alt: "3")[$3$] equal bunches. Find the number of balloons in each bunch, #math.equation(block: false, alt: "b ,")[$b ,$] by solving the equation #math.equation(block: false, alt: "3 b equals 18 .")[$3 b = 18 .$]

#strong[Teaching] Connie’s kindergarten class has #math.equation(block: false, alt: "24")[$24$] children. She wants them to get into #math.equation(block: false, alt: "4")[$4$] equal groups. Find the number of children in each group, #math.equation(block: false, alt: "g ,")[$g ,$] by solving the equation #math.equation(block: false, alt: "4 g equals 24 .")[$4 g = 24 .$]

#solutionbox[
6 children
]

#strong[Ticket price] Daria paid #math.equation(block: false, alt: "$36.25")[$"$36.25"$] for #math.equation(block: false, alt: "5")[$5$] children’s tickets at the ice skating rink. Find the price of each ticket, #math.equation(block: false, alt: "p ,")[$p ,$] by solving the equation #math.equation(block: false, alt: "5 p equals 36.25 .")[$5 p = 36.25 .$]

#strong[Unit price] Nishant paid #math.equation(block: false, alt: "$12.96")[$"$12.96"$] for a pack of #math.equation(block: false, alt: "12")[$12$] juice bottles. Find the price of each bottle, #math.equation(block: false, alt: "b ,")[$b ,$] by solving the equation #math.equation(block: false, alt: "12 b equals 12.96 .")[$12 b = 12.96 .$]

#solutionbox[
\$1.08
]

#strong[Fuel economy] Tania’s SUV gets half as many miles per gallon (mpg) as her husband’s hybrid car. The SUV gets #math.equation(block: false, alt: "18 mpg .")[$"18 mpg" .$] Find the miles per gallons, #math.equation(block: false, alt: "m ,")[$m ,$] of the hybrid car, by solving the equation #math.equation(block: false, alt: "the fraction 1 over 2 m equals 18 .")[$frac(1, 2) #h(0.1em) m = 18 .$]

#strong[Fabric] The drill team used #math.equation(block: false, alt: "14")[$14$] yards of fabric to make flags for one-third of the members. Find how much fabric, #math.equation(block: false, alt: "f ,")[$f ,$] they would need to make flags for the whole team by solving the equation #math.equation(block: false, alt: "the fraction 1 over 3 f equals 14 .")[$frac(1, 3) #h(0.1em) f = 14 .$]

#solutionbox[
42 yards
]


==== Writing Exercises

Frida started to solve the equation #math.equation(block: false, alt: "−3 x equals 36")[$−3 x = 36$] by adding #math.equation(block: false, alt: "3")[$3$] to both sides. Explain why Frida’s method will result in the correct solution.

Emiliano thinks #math.equation(block: false, alt: "x equals 40")[$x = 40$] is the solution to the equation #math.equation(block: false, alt: "the fraction 1 over 2 x equals 80 .")[$frac(1, 2) #h(0.1em) x = 80 .$] Explain why he is wrong.

#solutionbox[
Answer will vary.
]


==== Self Check

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

#figure(figph[A self-assessment chart for math skills, asking students to rate their ability to solve equations using division/multiplication properties and simplifying them, across three confidence levels.], alt: "A self-assessment chart for math skills, asking students to rate their ability to solve equations using division/multiplication properties and simplifying them, across three confidence levels.", caption: none)

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?