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== 2.2#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"))[
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

You may have noticed that all of the equations we have solved so far have been of the form #math.equation(block: false, alt: "x plus a equals b")[$x + a = b$] or #math.equation(block: false, alt: "x minus a equals b")[$x − a = b$]. We were able to isolate the variable by adding or subtracting the constant term on the side of the equation with the variable. Now we will see how to solve equations that have a variable multiplied by a constant and so will require division to isolate the variable.

Let’s look at our puzzle again with the envelopes and counters in .

#figure(figph[This image illustrates a workspace divided into two sides. The content of the left side is equal to the content of the right side. On the left side, there are two envelopes each containing an unknown but equal number of counters. On the right side are six counters.], alt: "This image illustrates a workspace divided into two sides. The content of the left side is equal to the content of the right side. On the left side, there are two envelopes each containing an unknown but equal number of counters. On the right side are six counters.", caption: [The illustration shows a model of an equation with one variable multiplied by a constant. On the left side of the workspace are two instances of the unknown (envelope), while on the right side of the workspace are six counters.])

In the illustration there are two identical envelopes that contain the same number of counters. Remember, the left side of the workspace must equal the right side, but the counters on the left side are “hidden” in the envelopes. So how many counters are in each envelope?

How do we determine the number? We have to separate the counters on the right side into two groups of the same size to correspond with the two envelopes on the left side. The 6 counters divided into 2 equal groups gives 3 counters in each group (since #math.equation(block: false, alt: "6 divided by 2 equals 3")[$6 ÷ 2 = 3$]).

What equation models the situation shown in ? There are two envelopes, and each contains #math.equation(block: false, alt: "x")[$x$] counters. Together, the two envelopes must contain a total of 6 counters.

#figure(figph[This image illustrates a workspace divided into two sides. The content of the left side is equal to the content of the right side. On the left side, there are two envelopes each containing an unknown but equal number of counters. On the right side are six counters. Underneath the image is the equation modeled by the counters: 2 x equals 6.], alt: "This image illustrates a workspace divided into two sides. The content of the left side is equal to the content of the right side. On the left side, there are two envelopes each containing an unknown but equal number of counters. On the right side are six counters. Underneath the image is the equation modeled by the counters: 2 x equals 6.", caption: [The illustration shows a model of the equation #math.equation(block: false, alt: "2 x equals 6")[$2 x = 6$].])

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The mathematical equation '2x = 6' is displayed in a simple, clear font against a white background.], alt: "The mathematical equation '2x = 6' is displayed in a simple, clear font against a white background.", caption: none)]),
  [If we divide both sides of the equation by 2, as we did with the envelopes and counters,], [#figure(figph[An image demonstrating the final step in solving a linear equation, showing 2x divided by 2 equals 6 divided by 2, which simplifies to x=3.], alt: "An image demonstrating the final step in solving a linear equation, showing 2x divided by 2 equals 6 divided by 2, which simplifies to x=3.", caption: none)],
  [we get:], [#figure(figph[The image shows a simple mathematical expression: x=3.], alt: "The image shows a simple mathematical expression: x=3.", caption: none)],
))

We found that each envelope contains 3 counters. Does this check? We know #math.equation(block: false, alt: "2 times 3 equals 6")[$2 · 3 = 6$], so it works! Three counters in each of two envelopes does equal six!

This example leads to the #strong[Division Property of Equality].

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[The Division Property of Equality]

For any numbers #emph[a], #emph[b], and #emph[c], and #math.equation(block: false, alt: "c not equal to 0")[$c ≠ 0$],

#math.equation(block: true, alt: "If, a, equals, b ,; then, the fraction a over c, equals, the fraction b over c")[$"If" &  & a & = & b , \ "then" &  & frac(a, c) & = & frac(b, c)$]When you divide both sides of an equation by any non-zero number, you still have equality.
]

The goal in solving an equation is to ‘undo’ the operation on the variable. In the next example, the variable is multiplied by 5, so we will divide both sides by 5 to ‘undo’ the multiplication.

#examplebox("Example 1")[][
Solve: #math.equation(block: false, alt: "5 x equals −27 .")[$5 x = −27 .$]

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [To isolate #math.equation(block: false, alt: "x")[$x$], “undo” the multiplication by 5.], [#figure(figph[A mathematical equation is displayed with a white background and dark gray text, showing '5x = -27'.], alt: "A mathematical equation is displayed with a white background and dark gray text, showing '5x = -27'.", caption: none)], [],
  [Divide to ‘undo’ the multiplication.], [#figure(figph[A mathematical equation shows '5x divided by 5 equals -27 divided by 5'. The number 5 in the denominators on both sides of the equation is highlighted in red.], alt: "A mathematical equation shows '5x divided by 5 equals -27 divided by 5'. The number 5 in the denominators on both sides of the equation is highlighted in red.", caption: none)], [],
  [Simplify.], [#figure(figph[The image shows a mathematical equation in black text on a white background, displaying x = -27/5, indicating that x is equal to negative twenty-seven over five.], alt: "The image shows a mathematical equation in black text on a white background, displaying x = -27/5, indicating that x is equal to negative twenty-seven over five.", caption: none)], [],
  [Check:], [#figure(figph[A mathematical equation shows '5x = -27' in black text on a white background. This is a linear equation with one variable, where 5 times x equals negative 27.], alt: "A mathematical equation shows '5x = -27' in black text on a white background. This is a linear equation with one variable, where 5 times x equals negative 27.", caption: none)], [],
  [Substitute #math.equation(block: false, alt: "minus the fraction 27 over 5")[$− frac(27, 5)$] for #math.equation(block: false, alt: "x .")[$x .$]], [#figure(figph[This image displays the equation 5(-27/5) =? -27, a math problem testing basic multiplication with fractions and negative numbers. The statement is true as 5 \* (-27/5) simplifies to -27.], alt: "This image displays the equation 5(-27/5) =? -27, a math problem testing basic multiplication with fractions and negative numbers. The statement is true as 5 * (-27/5) simplifies to -27.", caption: none)], [],
  [], [#figure(figph[The equation -27 = -27 is displayed with a checkmark, indicating that the statement is correct.], alt: "The equation -27 = -27 is displayed with a checkmark, indicating that the statement is correct.", caption: none)], [],
  [], [], [Since this is a true statement, #math.equation(block: false, alt: "x equals minus the fraction 27 over 5")[$x = − frac(27, 5)$] #linebreak() is the solution to #math.equation(block: false, alt: "5 x equals −27")[$5 x = −27$].],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "3 y equals −41 .")[$3 y = −41 .$]

#solutionbox[
#math.equation(block: true, alt: "y equals the fraction −41 over 3")[$y = frac(−41, 3)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "4 z equals −55 .")[$4 z = −55 .$]

#solutionbox[
#math.equation(block: true, alt: "z equals the fraction −55 over 4")[$z = frac(−55, 4)$]
]
]

Consider the equation #math.equation(block: false, alt: "the fraction x over 4 equals 3")[$frac(x, 4) = 3$]. We want to know what number divided by 4 gives 3. So to “undo” the division, we will need to multiply by 4. The #strong[Multiplication Property of Equality] will allow us to do this. This property says that if we start with two equal quantities and multiply both by the same number, the results are equal.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[The Multiplication Property of Equality]

For any numbers #emph[a], #emph[b], and #emph[c],

#math.equation(block: true, alt: "If, a, equals, b ,; then, a c, equals, b c")[$"If" &  & a & = & b , \ "then" &  & a c & = & b c$]If you multiply both sides of an equation by the same number, you still have equality.
]

#examplebox("Example 2")[][
Solve: #math.equation(block: false, alt: "the fraction y over −7 equals −14 .")[$frac(y, −7) = −14 .$]

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

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[A mathematical equation is displayed on a white background: y divided by -7 equals -14.], alt: "A mathematical equation is displayed on a white background: y divided by -7 equals -14.", caption: none)], [],
  [Multiply both sides by #math.equation(block: false, alt: "−7")[$−7$].], [#figure(figph[A mathematical equation shows -7 multiplied by the fraction y over -7, which is equal to -7 multiplied by -14. The -7 values on both sides of the equation are highlighted in red.], alt: "A mathematical equation shows -7 multiplied by the fraction y over -7, which is equal to -7 multiplied by -14. The -7 values on both sides of the equation are highlighted in red.", caption: none)], [],
  [Multiply.], [#figure(figph[A mathematical equation is displayed on a white background: -7y divided by 7 equals 98.], alt: "A mathematical equation is displayed on a white background: -7y divided by 7 equals 98.", caption: none)], [],
  [Simplify.], [#figure(figph[The image displays a simple mathematical equation, 'y = 98,' on a plain white background, indicating that the variable 'y' has a fixed value of ninety-eight.], alt: "The image displays a simple mathematical equation, 'y = 98,' on a plain white background, indicating that the variable 'y' has a fixed value of ninety-eight.", caption: none)], [],
  [Check: #math.equation(block: false, alt: "the fraction y over −7 equals −14")[$frac(y, −7) = −14$]], [], [],
  [Substitute #math.equation(block: false, alt: "y equals 98")[$y = 98$].], [#figure(figph[A mathematical equation showing '98 divided by -7 equals ? equals -14'. The red number 98 is in the numerator, and -7 is in the denominator. The question mark is above the first equals sign, implying a check of the equality.], alt: "A mathematical equation showing '98 divided by -7 equals ? equals -14'. The red number 98 is in the numerator, and -7 is in the denominator. The question mark is above the first equals sign, implying a check of the equality.", caption: none)], [],
  [Divide.], [#figure(figph[The image displays the mathematical equation -14 = -14, followed by a checkmark, indicating that the equation is correct.], alt: "The image displays the mathematical equation -14 = -14, followed by a checkmark, indicating that the equation is correct.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the fraction a over −7 equals −42 .")[$frac(a, −7) = −42 .$]

#solutionbox[
#math.equation(block: true, alt: "a equals 294")[$a = 294$]
]
]

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

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

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

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[The image displays the simple algebraic equation '-n = 9' in black text against a white background.], alt: "The image displays the simple algebraic equation '-n = 9' in black text against a white background.", caption: none)], [],
  [Remember #math.equation(block: false, alt: "− n")[$− n$] is equivalent to #math.equation(block: false, alt: "−1 n")[$−1 n$].], [#figure(figph[The image displays the equation -1n = 9 in black text on a white background.], alt: "The image displays the equation -1n = 9 in black text on a white background.", caption: none)], [],
  [Divide both sides by #math.equation(block: false, alt: "−1")[$−1$].], [#figure(figph[A mathematical equation illustrating the division of both sides by -1: -1n / -1 = 9 / -1. This step is typically used to isolate the variable 'n'.], alt: "A mathematical equation illustrating the division of both sides by -1: -1n / -1 = 9 / -1. This step is typically used to isolate the variable 'n'.", caption: none)], [],
  [Divide.], [#figure(figph[The mathematical expression 'n = -9' is displayed on a white background.], alt: "The mathematical expression 'n = -9' is displayed on a white background.", caption: none)], [],
  [Notice that there are two other ways to solve #math.equation(block: false, alt: "− n equals 9")[$− n = 9$]. We can also solve this equation by multiplying both sides by #math.equation(block: false, alt: "−1")[$−1$] and also by taking the opposite of both sides.], [], [],
  [Check:], [#figure(figph[The mathematical equation '-n = 9' is displayed in white text on a white background, suggesting a calculation or problem. The characters are clearly visible.], alt: "The mathematical equation '-n = 9' is displayed in white text on a white background, suggesting a calculation or problem. The characters are clearly visible.", caption: none)], [],
  [Substitute #math.equation(block: false, alt: "n equals −9")[$n = −9$].], [#figure(figph[The image displays the mathematical expression '-(-9) =? 9', posing a question about whether negative negative nine equals nine. Since the two negative signs cancel each other out, -(-9) simplifies to 9, confirming the equality.], alt: "The image displays the mathematical expression '-(-9) =? 9', posing a question about whether negative negative nine equals nine. Since the two negative signs cancel each other out, -(-9) simplifies to 9, confirming the equality.", caption: none)], [],
  [Simplify.], [#figure(figph[The equation 9=9 with a checkmark, symbolizing a correct or confirmed statement.], alt: "The equation 9=9 with a checkmark, symbolizing a correct or confirmed statement.", caption: none)], [],
))
]
]

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

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

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

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

#examplebox("Example 4")[][
Solve: #math.equation(block: false, alt: "the fraction 3 over 4 x equals 12 .")[$frac(3, 4) x = 12 .$]

#solutionbox[
Since the product of a number and its reciprocal is 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 3 over 4")[$frac(3, 4)$].

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[A mathematical equation showing three-fourths multiplied by x equals 12, expressed as '3/4 x = 12' on a white background.], alt: "A mathematical equation showing three-fourths multiplied by x equals 12, expressed as '3/4 x = 12' on a white background.", caption: none)], [],
  [Multiply by the reciprocal of #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$].], [#figure(figph[An algebraic equation showing the step to solve for x by multiplying both sides of (3/4)x=12 by the reciprocal 4/3.], alt: "An algebraic equation showing the step to solve for x by multiplying both sides of (3/4)x=12 by the reciprocal 4/3.", caption: none)], [],
  [Reciprocals multiply to 1.], [#figure(figph[A mathematical equation shows '1x = 4/3 multiplied by 12/1'.], alt: "A mathematical equation shows '1x = 4/3 multiplied by 12/1'.", caption: none)], [],
  [Multiply.], [#figure(figph[The image displays a simple mathematical equation, 'x = 16', rendered in a clear, dark gray typeface against a plain white background.], alt: "The image displays a simple mathematical equation, 'x = 16', rendered in a clear, dark gray typeface against a plain white background.", caption: none)], [],
  [Notice that we could have divided both sides of the equation #math.equation(block: false, alt: "the fraction 3 over 4 x equals 12")[$frac(3, 4) x = 12$] by  #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$] to isolate #math.equation(block: false, alt: "x")[$x$]. While this would work, most people would find multiplying by the reciprocal easier.], [], [],
  [Check:], [#figure(figph[A mathematical equation is displayed, showing '3/4x = 12' on a white background.], alt: "A mathematical equation is displayed, showing '3/4x = 12' on a white background.", caption: none)], [],
  [Substitute #math.equation(block: false, alt: "x equals 16")[$x = 16$].], [#figure(figph[A mathematical expression (3/4) \* 16 =? 12, challenging the viewer to determine if the equality is true. The '16' is highlighted in red, and a question mark sits above the equals sign.], alt: "A mathematical expression (3/4) * 16 =? 12, challenging the viewer to determine if the equality is true. The '16' is highlighted in red, and a question mark sits above the equals sign.", caption: none)], [],
  [], [#figure(figph[The equation '12 = 12' is presented, accompanied by a checkmark confirming its accuracy.], alt: "The equation '12 = 12' is presented, accompanied by a checkmark confirming its accuracy.", caption: none)], [],
))
]
]

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

#solutionbox[
#math.equation(block: true, alt: "n equals 35")[$n = 35$]
]
]

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

#solutionbox[
#math.equation(block: true, alt: "y equals 18")[$y = 18$]
]
]

In the next example, all the variable terms are on the right side of the equation. As always, our goal in solving the equation is to isolate the variable.

#examplebox("Example 5")[][
Solve: #math.equation(block: false, alt: "the fraction 8 over 15 equals minus the fraction 4 over 5 x .")[$frac(8, 15) = − frac(4, 5) x .$]

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[A mathematical equation is displayed on a white background, which reads '8 over 15 equals negative 4 over 5 x'.], alt: "A mathematical equation is displayed on a white background, which reads '8 over 15 equals negative 4 over 5 x'.", caption: none)], [],
  [Multiply by the reciprocal of #math.equation(block: false, alt: "minus the fraction 4 over 5")[$− frac(4, 5)$].], [#figure(figph[A mathematical equation is displayed, showing the product of two fractions, (-5/4) and (8/15), on the left side, equated to the product of (-5/4) and (-4/5x) on the right.], alt: "A mathematical equation is displayed, showing the product of two fractions, (-5/4) and (8/15), on the left side, equated to the product of (-5/4) and (-4/5x) on the right.", caption: none)], [],
  [Reciprocals multiply to 1.], [#figure(figph[A mathematical equation shows the simplification of a fraction: -(5\*4\*2)/(4\*3\*5) = 1x. Common factors '5' and '4' are crossed out from both the numerator and the denominator, leading to further calculation.], alt: "A mathematical equation shows the simplification of a fraction: -(5*4*2)/(4*3*5) = 1x. Common factors '5' and '4' are crossed out from both the numerator and the denominator, leading to further calculation.", caption: none)], [],
  [Multiply.], [#figure(figph[A mathematical equation shows a negative fraction, '-2/3,' which is set equal to the variable 'x.' The expression is centered against a plain white background.], alt: "A mathematical equation shows a negative fraction, '-2/3,' which is set equal to the variable 'x.' The expression is centered against a plain white background.", caption: none)], [],
  [Check:], [#figure(figph[A mathematical equation is displayed on a white background, which reads '8/15 = -4/5x'], alt: "A mathematical equation is displayed on a white background, which reads '8/15 = -4/5x'", caption: none)], [],
  [Let #math.equation(block: false, alt: "x equals minus the fraction 2 over 3")[$x = − frac(2, 3)$].], [#figure(figph[A mathematical equation showing the fraction 8/15 equals the product of -4/5 and -2/3. The numbers 2 and 3 in the second fraction are highlighted in red.], alt: "A mathematical equation showing the fraction 8/15 equals the product of -4/5 and -2/3. The numbers 2 and 3 in the second fraction are highlighted in red.", caption: none)], [],
  [], [#figure(figph[A mathematical equation showing the fraction 8 over 15 is equal to 8 over 15, followed by a checkmark indicating correctness.], alt: "A mathematical equation showing the fraction 8 over 15 is equal to 8 over 15, followed by a checkmark indicating correctness.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the fraction 9 over 25 equals minus the fraction 4 over 5 z .")[$frac(9, 25) = − frac(4, 5) #h(0.2em) z .$]

#solutionbox[
#math.equation(block: true, alt: "z equals minus the fraction 9 over 20")[$z = − frac(9, 20)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the fraction 5 over 6 equals minus the fraction 8 over 3 r .")[$frac(5, 6) = − frac(8, 3) #h(0.2em) r .$]

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


=== Solve Equations That Require Simplification

Many equations start out more complicated than the ones we have been working with.

With these more complicated equations the first step is to simplify both sides of the equation as much as possible. This usually involves combining like terms or using the distributive property.

#examplebox("Example 6")[][
Solve: #math.equation(block: false, alt: "14 minus 23 equals 12 y minus 4 y minus 5 y .")[$14 − 23 = 12 y − 4 y − 5 y .$]

#solutionbox[
Begin by simplifying each side of the equation.

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[A mathematical equation is displayed on a white background: 14 - 23 = 12y - 4y - 5y.], alt: "A mathematical equation is displayed on a white background: 14 - 23 = 12y - 4y - 5y.", caption: none)], [],
  [Simplify each side.], [#figure(figph[A mathematical equation shows -9 = 3y, depicting a linear algebraic expression to solve for the variable 'y'.], alt: "A mathematical equation shows -9 = 3y, depicting a linear algebraic expression to solve for the variable 'y'.", caption: none)], [],
  [Divide both sides by #math.equation(block: false, alt: "3")[$3$].], [#figure(figph[A mathematical equation shows '-3 = y' on a white background, representing that the variable y is equal to negative three.], alt: "A mathematical equation shows '-3 = y' on a white background, representing that the variable y is equal to negative three.", caption: none)], [],
  [Check:], [#figure(figph[A mathematical equation is displayed, showing '14 - 23 = 12y - 4y - 5y' in a simple black font against a white background.], alt: "A mathematical equation is displayed, showing '14 - 23 = 12y - 4y - 5y' in a simple black font against a white background.", caption: none)], [],
  [Substitute #math.equation(block: false, alt: "y equals −3")[$y = −3$].], [#figure(figph[A mathematical equation showing 14 minus 23 on the left side, and 12 times -3 minus 4 times -3 minus 5 times -3 on the right side. The equation reads: 14 - 23 = 12(-3) - 4(-3) - 5(-3).], alt: "A mathematical equation showing 14 minus 23 on the left side, and 12 times -3 minus 4 times -3 minus 5 times -3 on the right side. The equation reads: 14 - 23 = 12(-3) - 4(-3) - 5(-3).", caption: none)], [],
  [], [#figure(figph[A mathematical equation shows '-9 = -36 + 12 + 15' centered on a white background. The numbers and symbols are displayed in a gray font.], alt: "A mathematical equation shows '-9 = -36 + 12 + 15' centered on a white background. The numbers and symbols are displayed in a gray font.", caption: none)], [],
  [], [#figure(figph[A simple mathematical equation '-9 = -9' is displayed, followed by a checkmark, indicating its correctness.], alt: "A simple mathematical equation '-9 = -9' is displayed, followed by a checkmark, indicating its correctness.", caption: none)], [],
))
]
]

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

#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: "−4 open parenthesis a minus 3 close parenthesis minus 7 equals 25 .")[$−4 ( a − 3 ) − 7 = 25 .$]

#solutionbox[
Here we will simplify each side of the equation by using the distributive property first.

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#figure(figph[A mathematical equation is displayed, showing -4 multiplied by the quantity (a minus 3), then minus 7, which equals 25.], alt: "A mathematical equation is displayed, showing -4 multiplied by the quantity (a minus 3), then minus 7, which equals 25.", caption: none)], [],
  [Distribute.], [#figure(figph[A mathematical equation is presented, reading as '-4a + 12 - 7 = 25' against a white background.], alt: "A mathematical equation is presented, reading as '-4a + 12 - 7 = 25' against a white background.", caption: none)], [],
  [Simplify.], [#figure(figph[A mathematical equation is displayed on a white background: -4a + 5 = 25.], alt: "A mathematical equation is displayed on a white background: -4a + 5 = 25.", caption: none)], [],
  [Simplify.], [#figure(figph[The image shows a mathematical equation in black text on a white background, which reads '-4q = 20'.], alt: "The image shows a mathematical equation in black text on a white background, which reads '-4q = 20'.", caption: none)], [],
  [Divide both sides by #math.equation(block: false, alt: "−4")[$−4$] to isolate #math.equation(block: false, alt: "a")[$a$].], [#figure(figph[A mathematical equation shows '-4a over -4 equals 20 over -4.' The negative four in the denominator on both sides is highlighted in red, indicating division by negative four to solve for 'a'.], alt: "A mathematical equation shows '-4a over -4 equals 20 over -4.' The negative four in the denominator on both sides is highlighted in red, indicating division by negative four to solve for 'a'.", caption: none)], [],
  [Divide.], [#figure(figph[The image displays the equation 'a = -5' in a clear, black font against a white background.], alt: "The image displays the equation 'a = -5' in a clear, black font against a white background.", caption: none)], [],
  [Check:], [#figure(figph[A mathematical equation is displayed, which reads as -4(a - 3) - 7 = 25. This equation involves an unknown variable 'a' and requires algebraic manipulation to solve for 'a'.], alt: "A mathematical equation is displayed, which reads as -4(a - 3) - 7 = 25. This equation involves an unknown variable 'a' and requires algebraic manipulation to solve for 'a'.", caption: none)], [],
  [Substitute #math.equation(block: false, alt: "a equals −5")[$a = −5$].], [#figure(figph[A mathematical equation is shown, asking to verify if -4(-5 - 3) - 7 equals 25. The number -5 is highlighted in red within the parentheses.], alt: "A mathematical equation is shown, asking to verify if -4(-5 - 3) - 7 equals 25. The number -5 is highlighted in red within the parentheses.", caption: none)], [],
  [], [#figure(figph[A mathematical equation is displayed, showing '-4(-8) - 7' on the left side and '25' on the right, with a question mark above the equals sign, indicating a check for equality.], alt: "A mathematical equation is displayed, showing '-4(-8) - 7' on the left side and '25' on the right, with a question mark above the equals sign, indicating a check for equality.", caption: none)], [],
  [], [#figure(figph[A mathematical equation is displayed as 32 - 7 ?= 25, with a question mark positioned above the equals sign, indicating a query about the truth of the statement. The equation is correct as 32 minus 7 equals 25.], alt: "A mathematical equation is displayed as 32 - 7 ?= 25, with a question mark positioned above the equals sign, indicating a query about the truth of the statement. The equation is correct as 32 minus 7 equals 25.", caption: none)], [],
  [], [#figure(figph[A simple equation '25 = 25' with a checkmark, indicating correctness or completion. The numbers are bold and clear against a white background, highlighting a straightforward mathematical verification.], alt: "A simple equation '25 = 25' with a checkmark, indicating correctness or completion. The numbers are bold and clear against a white background, highlighting a straightforward mathematical verification.", caption: none)], [],
))
]
]

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

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

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

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

Now we have covered all four properties of equality—subtraction, addition, division, and multiplication. We’ll list them all together here for easy reference.

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

#math.equation(block: true, alt: "Subtraction Property of Equality, Addition Property of Equality; For any real numbers a , b , and c ,, For any real numbers a , b , and c ,; if, a, equals, b ,; then, a minus c, equals, b minus c ., if, a, equals, b ,; then, a plus c, equals, b plus c .; Division Property of Equality, Multiplication Property of Equality; For any numbers a , b , and c , and c not equal to 0 ,, For any numbers a , b , and c ,; if, a, equals, b ,; then, the fraction a over c, equals, the fraction b over c ., if, a, equals, b ,; then, a c, equals, b c .")[$"Subtraction Property of Equality" &  & "Addition Property of Equality" \ "For any real numbers" #h(0.2em) a , b , "and" #h(0.2em) c , &  & "For any real numbers" #h(0.2em) a , b , "and" #h(0.2em) c , \ #h(1em) "if" & a & = & b , \ "then" & a − c & = & b − c . &  & #h(1em) "if" & a & = & b , \ "then" & a + c & = & b + c . \ "Division Property of Equality" &  & "Multiplication Property of Equality" \ "For any numbers" #h(0.2em) a , b , "and" #h(0.2em) c , "and" #h(0.2em) c ≠ 0 , &  & "For any numbers" #h(0.2em) a , b , "and" #h(0.2em) c , \ #h(1em) "if" & a & = & b , \ "then" & frac(a, c) & = & frac(b, c) . &  & #h(1em) "if" & a & = & b , \ "then" & a c & = & b c .$]When you add, subtract, multiply, or divide the same quantity from both sides of an equation, you still have equality.
]


=== Translate to an Equation and Solve

In the next few examples, we will translate sentences into equations and then solve the equations. You might want to review the translation table in the previous chapter.

#examplebox("Example 8")[][
Translate and solve: The number 143 is the product of #math.equation(block: false, alt: "−11")[$−11$] and #emph[y].

#solutionbox[
Begin by translating the sentence into an equation.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Translate.], [#figure(figph[This image illustrates how to translate the word problem 'The number 143 is the product of -11 and y' into the algebraic equation '143 = -11y', with visual cues connecting the text to the equation.], alt: "This image illustrates how to translate the word problem 'The number 143 is the product of -11 and y' into the algebraic equation '143 = -11y', with visual cues connecting the text to the equation.", caption: none)]),
  [Divide by #math.equation(block: false, alt: "−11")[$−11$].], [#figure(figph[A mathematical equation shows both sides being divided by -11. The left side is 143/-11 and the right side is -11y/-11. The number -11 is in red font in the denominators.], alt: "A mathematical equation shows both sides being divided by -11. The left side is 143/-11 and the right side is -11y/-11. The number -11 is in red font in the denominators.", caption: none)],
  [Simplify.], [#figure(figph[The image shows a simple algebraic equation, '-13 = y', indicating that the variable 'y' is equal to negative thirteen.], alt: "The image shows a simple algebraic equation, '-13 = y', indicating that the variable 'y' is equal to negative thirteen.", caption: none)],
  [Check: #linebreak() 
       #math.equation(block: false, alt: "143, equals, −11 y; 143, equals ?, −11 open parenthesis −13 close parenthesis; 143, equals, 143 ✓")[$& 143 & = & −11 y \  & 143 & limits(=)^(?) & −11 ( −13 ) \  & 143 & = & 143 ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve: The number 132 is the product of #emph[−12] and #emph[y].

#solutionbox[
#math.equation(block: true, alt: "132 equals −12 y ; y equals −11")[$132 = −12 y ; y = −11$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve: The number 117 is the product of #emph[−13] and #emph[z].

#solutionbox[
#math.equation(block: true, alt: "117 equals −13 z ; z equals −9")[$117 = −13 z ; z = −9$]
]
]

#examplebox("Example 9")[][
Translate and solve: #math.equation(block: false, alt: "n")[$n$] divided by 8 is #math.equation(block: false, alt: "−32")[$−32$].

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [Begin by translating the sentence into an equation. #linebreak() Translate.], [#figure(figph[A mathematical statement and its corresponding equation are displayed. The statement reads 'n divided by 8 is -32.' Below it, the equation is written as 'n/8 = -32.'], alt: "A mathematical statement and its corresponding equation are displayed. The statement reads 'n divided by 8 is -32.' Below it, the equation is written as 'n/8 = -32.'", caption: none)], [],
  [Multiple both sides by 8.], [#figure(figph[A mathematical equation shows '8 multiplied by n divided by 8 equals 8 multiplied by negative 32.' The number 8 is highlighted in red on both sides of the equation.], alt: "A mathematical equation shows '8 multiplied by n divided by 8 equals 8 multiplied by negative 32.' The number 8 is highlighted in red on both sides of the equation.", caption: none)], [],
  [Simplify.], [#figure(figph[The text 'n = -256' is prominently displayed on a white background, indicating a numerical value or a mathematical expression.], alt: "The text 'n = -256' is prominently displayed on a white background, indicating a numerical value or a mathematical expression.", caption: none)], [],
  [Check:], [Is #math.equation(block: false, alt: "n")[$n$] divided by 8 equal to −32?], [],
  [Let #math.equation(block: false, alt: "n equals −256")[$n = −256$].], [Is #math.equation(block: false, alt: "−256")[$−256$] divided by #math.equation(block: false, alt: "8")[$8$] equal to #math.equation(block: false, alt: "−32")[$−32$]?], [],
  [Translate.], [#math.equation(block: false, alt: "the fraction −256 over 8 equals ? −32")[$frac(−256, 8) limits(=)^(?) −32$]], [],
  [Simplify.], [#math.equation(block: false, alt: "−32 equals −32 ✓")[$#h(0.6em) −32 = −32 ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve: #math.equation(block: false, alt: "n")[$n$] divided by 7 is equal to #math.equation(block: false, alt: "−21")[$−21$].

#solutionbox[
#math.equation(block: true, alt: "the fraction n over 7 equals −21 ; n equals −147")[$frac(n, 7) = −21 ; n = −147$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve: #math.equation(block: false, alt: "n")[$n$] divided by 8 is equal to #math.equation(block: false, alt: "−56")[$−56$].

#solutionbox[
#math.equation(block: true, alt: "the fraction n over 8 equals −56 ; n equals −448")[$frac(n, 8) = −56 ; n = −448$]
]
]

#examplebox("Example 10")[][
Translate and solve: The quotient of #math.equation(block: false, alt: "y")[$y$] and #math.equation(block: false, alt: "−4")[$−4$] is #math.equation(block: false, alt: "68")[$68$].

#solutionbox[
Begin by translating the sentence into an equation.

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [Translate.], [#figure(figph[An image illustrating the translation of the phrase 'The quotient of y and -4 is 68' into the algebraic equation 'y/-4 = 68', highlighted by teal brackets.], alt: "An image illustrating the translation of the phrase 'The quotient of y and -4 is 68' into the algebraic equation 'y/-4 = 68', highlighted by teal brackets.", caption: none)], [],
  [Multiply both sides by #math.equation(block: false, alt: "−4")[$−4$].], [#figure(figph[A mathematical equation shows -4 multiplied by the fraction y over -4, which equals -4 multiplied by 68. The numbers -4 and 68 are in black, while the multiplication signs are implied by parentheses.], alt: "A mathematical equation shows -4 multiplied by the fraction y over -4, which equals -4 multiplied by 68. The numbers -4 and 68 are in black, while the multiplication signs are implied by parentheses.", caption: none)], [],
  [Simplify.], [#figure(figph[The equation y = -272 is displayed in the upper right portion of a white background.], alt: "The equation y = -272 is displayed in the upper right portion of a white background.", caption: none)], [],
  [Check:], [Is the quotient of #math.equation(block: false, alt: "y")[$y$] and #math.equation(block: false, alt: "−4")[$−4$] equal to #math.equation(block: false, alt: "68")[$68$]?], [],
  [Let #math.equation(block: false, alt: "y equals −272")[$y = −272$].], [Is the quotient of #math.equation(block: false, alt: "−272")[$−272$] and #math.equation(block: false, alt: "−4")[$−4$] equal to #math.equation(block: false, alt: "68")[$68$]?], [],
  [Translate.], [#math.equation(block: false, alt: "the fraction −272 over −4 equals ? 68")[$frac(−272, −4) limits(=)^(?) 68$]], [],
  [Simplify.], [#math.equation(block: false, alt: "68 equals 68 ✓")[$#h(1.3em) 68 = 68 ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve: The quotient of #math.equation(block: false, alt: "q")[$q$] and #math.equation(block: false, alt: "−8")[$−8$] is 72.

#solutionbox[
#math.equation(block: true, alt: "the fraction q over −8 equals 72 ; q equals −576")[$frac(q, −8) = 72 ; q = −576$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve: The quotient of #math.equation(block: false, alt: "p")[$p$] and #math.equation(block: false, alt: "−9")[$−9$] is 81.

#solutionbox[
#math.equation(block: true, alt: "the fraction p over −9 equals 81 ; p equals −729")[$frac(p, −9) = 81 ; p = −729$]
]
]

#examplebox("Example 11")[][
Translate and solve: Three-fourths of #math.equation(block: false, alt: "p")[$p$] is 18.

#solutionbox[
Begin by translating the sentence into an equation. Remember, “of” translates into multiplication.

#figure(table(
  columns: 3, align: left, inset: 6pt,
  [Translate.], [#figure(figph[The image demonstrates how to translate the verbal phrase 'Three-fourths of p is 18' into the algebraic equation '3/4p = 18'. It visually connects parts of the phrase to their mathematical equivalents.], alt: "The image demonstrates how to translate the verbal phrase 'Three-fourths of p is 18' into the algebraic equation '3/4p = 18'. It visually connects parts of the phrase to their mathematical equivalents.", caption: none)], [],
  [Multiply both sides by #math.equation(block: false, alt: "the fraction 4 over 3 .")[$frac(4, 3) .$]], [#figure(figph[A mathematical equation shows (4/3) multiplied by (3/4)p on the left side, equaling (4/3) multiplied by 18 on the right side. The numbers 4 and 3 are highlighted in red.], alt: "A mathematical equation shows (4/3) multiplied by (3/4)p on the left side, equaling (4/3) multiplied by 18 on the right side. The numbers 4 and 3 are highlighted in red.", caption: none)], [],
  [Simplify.], [#figure(figph[The image shows the mathematical expression 'p = 24' in black text against a plain white background.], alt: "The image shows the mathematical expression 'p = 24' in black text against a plain white background.", caption: none)], [],
  [Check:], [Is three-fourths of #emph[p] equal to 18?], [],
  [Let #math.equation(block: false, alt: "p equals 24 .")[$p = 24 .$]], [Is three-fourths of 24  equal to  18?], [],
  [Translate.], [#math.equation(block: false, alt: "the fraction 3 over 4 times 24 equals ? 18")[$frac(3, 4) · #h(0.2em) 24 #h(0.2em) limits(=)^(?) 18$]], [],
  [Simplify.], [#math.equation(block: false, alt: "18 equals 18 ✓")[$#h(1.6em) 18 = 18 ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve: Two-fifths of #math.equation(block: false, alt: "f")[$f$] is 16.

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 over 5 f equals 16 ; f equals 40")[$frac(2, 5) f = 16 ; f = 40$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve: Three-fourths of #math.equation(block: false, alt: "f")[$f$] is 21.

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 over 4 f equals 21 ; f equals 28")[$frac(3, 4) f = 21 ; f = 28$]
]
]

#examplebox("Example 12")[][
Translate and solve: The sum of three-eighths and #math.equation(block: false, alt: "x")[$x$] is one-half.

#solutionbox[
Begin by translating the sentence into an equation.

#figure(table(
  columns: 4, align: left, inset: 6pt,
  [Translate.], [#figure(figph[An image illustrating the translation of a word problem into a mathematical equation. The sentence 'The sum of three-eighths and x is 1/2' is shown, with brackets linking parts to the equation '3/8 + x = 1/2' below.], alt: "An image illustrating the translation of a word problem into a mathematical equation. The sentence 'The sum of three-eighths and x is 1/2' is shown, with brackets linking parts to the equation '3/8 + x = 1/2' below.", caption: none)], [], [],
  [Subtract #math.equation(block: false, alt: "the fraction 3 over 8")[$frac(3, 8)$] from each side.], [#figure(figph[A mathematical equation is displayed on a white background, reading '3/8 - 3/8 + x = 1/2 - 3/8'. The fractions on the right side of the equals sign are partially in red.], alt: "A mathematical equation is displayed on a white background, reading '3/8 - 3/8 + x = 1/2 - 3/8'. The fractions on the right side of the equals sign are partially in red.", caption: none)], [], [],
  [Simplify and rewrite fractions with common denominators.], [#figure(figph[A mathematical equation displays 'x = 4/8 - 3/8' on a white background, representing a subtraction problem with fractions sharing a common denominator.], alt: "A mathematical equation displays 'x = 4/8 - 3/8' on a white background, representing a subtraction problem with fractions sharing a common denominator.", caption: none)], [], [],
  [Simplify.], [], [], [#figure(figph[A mathematical equation on a white background displays 'x = 1/8'.], alt: "A mathematical equation on a white background displays 'x = 1/8'.", caption: none)],
  [Check:], [], [Is the sum of three-eighths and #math.equation(block: false, alt: "x")[$x$] equal to one-half?], [],
  [#math.equation(block: false, alt: "Let x equals the fraction 1 over 8 .")[$"Let" #h(0.2em) x = frac(1, 8) .$]], [], [Is the sum of three-eighths and one-eighth equal to one-half?], [],
  [Translate.], [], [#math.equation(block: false, alt: "the fraction 3 over 8 plus the fraction 1 over 8 equals ? the fraction 1 over 2")[$#h(0.2em) frac(3, 8) + frac(1, 8) limits(=)^(?) frac(1, 2)$]], [],
  [Simplify.], [], [#math.equation(block: false, alt: "the fraction 4 over 8 equals ? the fraction 1 over 2")[$#h(2em) frac(4, 8) limits(=)^(?) frac(1, 2)$]], [],
  [Simplify.], [], [#math.equation(block: false, alt: "the fraction 1 over 2 equals the fraction 1 over 2 ✓")[$#h(2em) frac(1, 2) = frac(1, 2) ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve: The sum of five-eighths and #emph[x] is one-fourth.

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 over 8 plus x equals the fraction 1 over 4 ; x equals minus the fraction 3 over 8")[$frac(5, 8) + x = frac(1, 4) ; x = − frac(3, 8)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve: The sum of three-fourths and #emph[x] is five-sixths.

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 over 4 plus x equals the fraction 5 over 6 ; x equals the fraction 1 over 12")[$frac(3, 4) + x = frac(5, 6) ; x = frac(1, 12)$]
]
]


=== Translate and Solve Applications

To solve applications using the Division and Multiplication Properties of Equality, we will follow the same steps we used in the last section. We will restate the problem in just one sentence, assign a variable, and then translate the sentence into an equation to solve.

#examplebox("Example 13")[][
Denae bought 6 pounds of grapes for \$10.74. What was the cost of one pound of grapes?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([What are you asked to find?], [The cost of 1 pound of grapes]),
  [Assign a variable.], [Let #math.equation(block: false, alt: "c")[$c$] = the cost of one pound.],
  [Write a sentence that gives the information to find it.], [The cost of 6 pounds is \$10.74.],
  [Translate into an equation.], [#math.equation(block: false, alt: "6 c equals 10.74")[$6 c = 10.74$]],
  [Solve.], [#math.equation(block: false, alt: "the fraction 6 c over 6 equals the fraction 10.74 over 6; c equals 1.79")[$frac(6 c, 6) = frac(10.74, 6) \ c = 1.79$]],
  [], [The grapes cost \$1.79 per pound.],
  [Check: If one pound costs \$1.79, do 6 pounds cost \#10.74?  #linebreak() 
    #math.equation(block: false, alt: "6 open parenthesis 1.79 close parenthesis, equals ?, 10.74; 10.74, equals, 10.74 ✓")[$6 ( 1.79 ) & limits(=)^(?) & 10.74 \ 10.74 & = & 10.74 ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve:

Arianna bought a 24-pack of water bottles for \$9.36. What was the cost of one water bottle?

#solutionbox[
\$0.39
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve:

At JB’s Bowling Alley, 6 people can play on one lane for \$34.98. What is the cost for each person?

#solutionbox[
\$5.83
]
]

#examplebox("Example 14")[][
Andreas bought a used car for \$12,000. Because the car was 4-years old, its price was #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$] of the original price, when the car was new. What was the original price of the car?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([What are you asked to find?], [The original price of the car]),
  [Assign a variable.], [Let #math.equation(block: false, alt: "p")[$p$] = the original price.],
  [Write a sentence that gives the information to find it.], [\$12,000 is #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$] of the original price.],
  [Translate into an equation.], [#math.equation(block: false, alt: "12,000 equals the fraction 3 over 4 p")[$12,000 = frac(3, 4) p$]],
  [Solve.], [#math.equation(block: false, alt: "the fraction 4 over 3 open parenthesis 12,000 close parenthesis equals the fraction 4 over 3 times the fraction 3 over 4 p; 16,000 equals p")[$frac(4, 3) ( 12,000 ) = frac(4, 3) · frac(3, 4) p \ 16,000 = p$]],
  [], [The original cost of the car was \$16,000.],
  [Check: Is #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$] of \$16,000 equal to \$12,000? #linebreak() 
    #math.equation(block: false, alt: "the fraction 3 over 4 times 16,000, equals ?, 12,000; 12,000, equals, 12,000 ✓")[$frac(3, 4) · 16,000 & limits(=)^(?) & 12,000 \ 12,000 & = & 12,000 ✓$]], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve:

The annual property tax on the Mehta’s house is \$1,800, calculated as #math.equation(block: false, alt: "the fraction 15 over 1,000")[$frac(15, 1","000)$] of the assessed value of the house. What is the assessed value of the Mehta’s house?

#solutionbox[
\$120,000
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Translate and solve:

Stella planted 14 flats of flowers in #math.equation(block: false, alt: "the fraction 2 over 3")[$frac(2, 3)$] of her garden. How many flats of flowers would she need to fill the whole garden?

#solutionbox[
21 flats
]
]


=== Key Concepts

- #strong[The Division Property of Equality]—For any numbers #emph[a], #emph[b], and #emph[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")[$frac(a, c) = frac(b, c)$]. #linebreak() 
When you divide both sides of an equation by any non-zero number, you still have equality.
- #strong[The Multiplication Property of Equality]—For any numbers #emph[a], #emph[b], and #emph[c], if #math.equation(block: false, alt: "a equals b")[$a = b$], then #math.equation(block: false, alt: "a c equals b c")[$a c = b c$]. #linebreak() 
If you multiply both sides of an equation by the same number, you still have equality.


==== Practice Makes Perfect

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

In the following exercises, solve each equation using the Division and Multiplication Properties of Equality and check the solution.

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

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

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

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

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

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

#math.equation(block: true, alt: "−809 equals 15 y")[$−809 = 15 y$]

#solutionbox[
#math.equation(block: true, alt: "y equals minus the fraction 809 over 15")[$y = − frac(809, 15)$]
]

#math.equation(block: true, alt: "−731 equals 19 y")[$−731 = 19 y$]

#math.equation(block: true, alt: "−37 p equals −541")[$−37 p = −541$]

#solutionbox[
#math.equation(block: true, alt: "p equals the fraction 541 over 37")[$p = frac(541, 37)$]
]

#math.equation(block: true, alt: "−19 m equals −586")[$−19 m = −586$]

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

#solutionbox[
#math.equation(block: true, alt: "z equals 13")[$z = 13$]
]

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

#math.equation(block: true, alt: "−13 x equals 0")[$−13 x = 0$]

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

#math.equation(block: true, alt: "24 x equals 0")[$24 x = 0$]

#math.equation(block: true, alt: "the fraction x over 4 equals 35")[$frac(x, 4) = 35$]

#solutionbox[
#math.equation(block: true, alt: "x equals 140")[$x = 140$]
]

#math.equation(block: true, alt: "the fraction z over 2 equals 54")[$frac(z, 2) = 54$]

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

#solutionbox[
#math.equation(block: true, alt: "q equals 100")[$q = 100$]
]

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

#math.equation(block: true, alt: "the fraction y over 9 equals −16")[$frac(y, 9) = −16$]

#solutionbox[
#math.equation(block: true, alt: "y equals −144")[$y = −144$]
]

#math.equation(block: true, alt: "the fraction q over 6 equals −38")[$frac(q, 6) = −38$]

#math.equation(block: true, alt: "the fraction m over −12 equals 45")[$frac(m, −12) = 45$]

#solutionbox[
#math.equation(block: true, alt: "m equals −540")[$m = −540$]
]

#math.equation(block: true, alt: "−24 equals the fraction p over −20")[$−24 = frac(p, −20)$]

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

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

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

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

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

#math.equation(block: true, alt: "− x equals −39")[$"−" x = −39$]

#math.equation(block: true, alt: "the fraction 2 over 3 y equals 48")[$frac(2, 3) y = 48$]

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

#math.equation(block: true, alt: "the fraction 3 over 5 r equals 75")[$frac(3, 5) r = 75$]

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

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

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

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

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

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

#math.equation(block: true, alt: "minus the fraction 7 over 10 x equals minus the fraction 14 over 3")[$− frac(7, 10) x = − frac(14, 3)$]

#solutionbox[
#math.equation(block: true, alt: "x equals the fraction 20 over 3")[$x = frac(20, 3)$]
]

#math.equation(block: true, alt: "the fraction 3 over 8 y equals minus the fraction 1 over 4")[$frac(3, 8) y = − frac(1, 4)$]

#math.equation(block: true, alt: "the fraction 7 over 12 equals minus the fraction 3 over 4 p")[$frac(7, 12) = − frac(3, 4) p$]

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

#math.equation(block: true, alt: "the fraction 11 over 18 equals minus the fraction 5 over 6 q")[$frac(11, 18) = − frac(5, 6) q$]

#math.equation(block: true, alt: "minus the fraction 5 over 18 equals minus the fraction 10 over 9 u")[$− frac(5, 18) = − frac(10, 9) u$]

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

#math.equation(block: true, alt: "minus the fraction 7 over 20 equals minus the fraction 7 over 4 v")[$− frac(7, 20) = − frac(7, 4) v$]

#strong[Solve Equations That Require Simplification]

In the following exercises, solve each equation requiring simplification.

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

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

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

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

#solutionbox[
#math.equation(block: true, alt: "n equals 88")[$n = 88$]
]

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

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

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

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

#math.equation(block: true, alt: "−10 open parenthesis q minus 4 close parenthesis minus 57 equals 93")[$−10 ( q − 4 ) − 57 = 93$]

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

#math.equation(block: true, alt: "−12 open parenthesis d minus 5 close parenthesis minus 29 equals 43")[$−12 ( d − 5 ) − 29 = 43$]

#math.equation(block: true, alt: "−10 open parenthesis x plus 4 close parenthesis minus 19 equals 85")[$−10 ( x + 4 ) − 19 = 85$]

#solutionbox[
#math.equation(block: true, alt: "x equals minus the fraction 72 over 5")[$x = − frac(72, 5)$]
]

#math.equation(block: true, alt: "−15 open parenthesis z plus 9 close parenthesis minus 11 equals 75")[$−15 ( z + 9 ) − 11 = 75$]

#strong[Mixed Practice]

In the following exercises, solve each equation.

#math.equation(block: true, alt: "the fraction 9 over 10 x equals 90")[$frac(9, 10) x = 90$]

#solutionbox[
#math.equation(block: true, alt: "x equals 100")[$x = 100$]
]

#math.equation(block: true, alt: "the fraction 5 over 12 y equals 60")[$frac(5, 12) y = 60$]

#math.equation(block: true, alt: "y plus 46 equals 55")[$y + 46 = 55$]

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

#math.equation(block: true, alt: "x plus 33 equals 41")[$x + 33 = 41$]

#math.equation(block: true, alt: "the fraction w over −2 equals 99")[$frac(w, −2) = 99$]

#solutionbox[
#math.equation(block: true, alt: "w equals −198")[$w = −198$]
]

#math.equation(block: true, alt: "the fraction s over −3 equals −60")[$frac(s, −3) = −60$]

#math.equation(block: true, alt: "27 equals 6 a")[$27 = 6 a$]

#solutionbox[
#math.equation(block: true, alt: "a equals the fraction 9 over 2")[$a = frac(9, 2)$]
]

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

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

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

#math.equation(block: true, alt: "z minus 16 equals −59")[$z − 16 = −59$]

#math.equation(block: true, alt: "m minus 41 equals −14")[$m − 41 = −14$]

#solutionbox[
#math.equation(block: true, alt: "m equals 27")[$m = 27$]
]

#math.equation(block: true, alt: "0.04 r equals 52.60")[$0.04 r = 52.60$]

#math.equation(block: true, alt: "63.90 equals 0.03 p")[$63.90 = 0.03 p$]

#solutionbox[
#math.equation(block: true, alt: "p equals 2130")[$p = 2130$]
]

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

#math.equation(block: true, alt: "84 equals −12 z")[$84 = −12 z$]

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

#math.equation(block: true, alt: "19.36 equals x minus 0.2 x")[$19.36 = x − 0.2 x$]

#math.equation(block: true, alt: "c minus 0.3 c equals 35.70")[$c − 0.3 c = 35.70$]

#solutionbox[
#math.equation(block: true, alt: "c equals 51")[$c = 51$]
]

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

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

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

#strong[Translate to an Equation and Solve]

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

187 is the product of #math.equation(block: false, alt: "−17")[$−17$] and #emph[m].

133 is the product of #math.equation(block: false, alt: "−19")[$−19$] and #emph[n].

#solutionbox[
#math.equation(block: true, alt: "133 equals −19 n ; n equals −7")[$133 = −19 n ; n = −7$]
]

#math.equation(block: false, alt: "−184")[$−184$] is the product of 23 and #emph[p].

#math.equation(block: false, alt: "−152")[$−152$] is the product of 8 and #emph[q].

#solutionbox[
#math.equation(block: true, alt: "−152 equals 8 q ; q equals −19")[$−152 = 8 q ; q = −19$]
]

#emph[u] divided by 7 is equal to #math.equation(block: false, alt: "−49")[$−49$].

#emph[r] divided by 12 is equal to #math.equation(block: false, alt: "−48")[$−48$].

#solutionbox[
#math.equation(block: true, alt: "the fraction r over 12 equals −48 ; r equals −576")[$frac(r, 12) = −48 ; r = −576$]
]

#emph[h] divided by #math.equation(block: false, alt: "−13")[$−13$] is equal to #math.equation(block: false, alt: "−65")[$−65$].

#emph[j] divided by #math.equation(block: false, alt: "−20")[$−20$] is equal to #math.equation(block: false, alt: "−80")[$−80$].

#solutionbox[
#math.equation(block: true, alt: "the fraction j over −20 equals −80 ; j equals 1,600")[$frac(j, −20) = −80 ; j = 1,600$]
]

The quotient #math.equation(block: false, alt: "c")[$c$] and #math.equation(block: false, alt: "−19")[$−19$] is 38.

The quotient of #math.equation(block: false, alt: "b")[$b$] and #math.equation(block: false, alt: "−6")[$−6$] is 18.

#solutionbox[
#math.equation(block: true, alt: "the fraction b over −6 equals 18 ; b equals −108")[$frac(b, −6) = 18 ; b = −108$]
]

The quotient of #math.equation(block: false, alt: "h")[$h$] and 26 is #math.equation(block: false, alt: "−52")[$−52$].

The quotient #math.equation(block: false, alt: "k")[$k$] and 22 is #math.equation(block: false, alt: "−66")[$−66$].

#solutionbox[
#math.equation(block: true, alt: "the fraction k over 22 equals −66 ; k equals −1,452")[$frac(k, 22) = −66 ; k = −1,452$]
]

Five-sixths of #emph[y] is 15.

Three-tenths of #emph[x] is 15.

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 over 10 x equals 15 ; x equals 50")[$frac(3, 10) x = 15 ; x = 50$]
]

Four-thirds of #emph[w] is 36.

Five-halves of #emph[v] is 50.

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 over 2 v equals 50 ; v equals 20")[$frac(5, 2) v = 50 ; v = 20$]
]

The sum of nine-tenths and #emph[g] is two-thirds.

The sum of two-fifths and #emph[f] is one-half.

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 over 5 plus f equals the fraction 1 over 2 ; f equals the fraction 1 over 10")[$frac(2, 5) + f = frac(1, 2) ; f = frac(1, 10)$]
]

The difference of #emph[p] and one-sixth is two-thirds.

The difference of #emph[q] and one-eighth is three-fourths.

#solutionbox[
#math.equation(block: true, alt: "q minus the fraction 1 over 8 equals the fraction 3 over 4 ; q equals the fraction 7 over 8")[$q − frac(1, 8) = frac(3, 4) ; q = frac(7, 8)$]
]

#strong[Translate and Solve Applications]

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

#strong[Kindergarten] Connie’s kindergarten class has 24 children. She wants them to get into 4 equal groups. How many children will she put in each group?

#strong[Balloons] Ramona bought 18 balloons for a party. She wants to make 3 equal bunches. How many balloons did she use in each bunch?

#solutionbox[
6 balloons
]

#strong[Tickets] Mollie paid \$36.25 for 5 movie tickets. What was the price of each ticket?

#strong[Shopping] Serena paid \$12.96 for a pack of 12 pairs of sport socks. What was the price of pair of sport socks?

#solutionbox[
\$1.08
]

#strong[Sewing] Nancy used 14 yards of fabric to make flags for one-third of the drill team. How much fabric, would Nancy need to make flags for the whole team?

#strong[MPG] John’s SUV gets 18 miles per gallon (mpg). This is half as many mpg as his wife’s hybrid car. How many miles per gallon does the hybrid car get?

#solutionbox[
36 mpg
]

#strong[Height] Aiden is 27 inches tall. He is #math.equation(block: false, alt: "the fraction 3 over 8")[$frac(3, 8)$] as tall as his father. How tall is his father?

#strong[Real estate] Bea earned \$11,700 commission for selling a house, calculated as #math.equation(block: false, alt: "the fraction 6 over 100")[$frac(6, 100)$] of the selling price. What was the selling price of the house?

#solutionbox[
\$195,000
]


==== Everyday Math

#strong[Commission] Every week Perry gets paid \$150 plus 12% of his total sales amount over \$1,250. Solve the equation #math.equation(block: false, alt: "840 equals 150 plus 0.12 open parenthesis a minus 1250 close parenthesis")[$840 = 150 + 0.12 ( a − 1250 )$] for #emph[a], to find the total amount Perry must sell in order to be paid \$840 one week.

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

#solutionbox[
15 49-cent stamps
]


==== Writing Exercises

Frida started to solve the equation #math.equation(block: false, alt: "−3 x equals 36")[$−3 x = 36$] by adding 3 to both sides. Explain why Frida’s method will not solve the equation.

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")[$frac(1, 2) x = 80$]. Explain why he is wrong.

#solutionbox[
Answers will vary.
]


==== Self Check

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

#figure(figph[This is a table that has five 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 “1) solve equations using the Division and Multiplication Properties of equality,” “2) solve equations that require simplification,” “3) translate to an equation and solve,” and “4) translate and solve applications.” The rest of the cells are blank.], alt: "This is a table that has five 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 “1) solve equations using the Division and Multiplication Properties of equality,” “2) solve equations that require simplification,” “3) translate to an equation and solve,” and “4) translate and solve applications.” 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?