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Single Variable Multiplication Equations

Solve one - step equations using multiplication.

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Single Variable Multiplication Equations

Let’s Think About It

Credit: Philip Leara
Source: https://www.flickr.com/photos/philleara/7043822869/in/photolist-bJrr8T-a7msSe-bAhHuS-iTdFC5-7btUB6-4GZHSm-hYWteT-f8ecTT-5maKq1-6yaZjW-i2EUcw-d9aAvh-qeYrLQ-6xarzP-4Ag76i-8y25gs-a7nKAA-gkatRL-fNbazq-5TRNiC-bKJAai-fPukaS-eqw2tK-fhgbhM-fNBznB-a8PtHq-hbT1br-a9fi3G-7KzBXp-8CMehR-k4iXhn-pX1crj-ABpTx-8n6vSf-8n6vPj-iDbfpP-aqr4XN-ehE6tk-gchYqd-dFLSzN-qeYrz7-gchZX1-bAhHEo-qxcWEa-ou2G9S-q2ugmY-6FeDgr-oZsVdU-iUuUVR-7qbtLT
License: CC BY-NC 3.0

Mr. Ricky’s Biology class is going to the botanical gardens to study the different kinds of flowers and plants that are there. They have raised 68 dollars for tickets so far. The student rate for each ticket is 5 dollars. Mr. Ricky asked the students to figure out how many tickets they can buy. How many tickets can they afford with 68 dollars?

In this concept, you will learn to solve single variable multiplication equations.

Guidance

In the algebraic equation below, the variable \begin{align*}z\end{align*}z represents a number.

\begin{align*}z \times 2 = 8\end{align*}

z×2=8

What number does \begin{align*}z\end{align*}z represent?

You can find out by asking yourself, “What number, when multiplied by 2, equals 8?”

Since \begin{align*}4 \times 2 = 8\end{align*}4×2=8\begin{align*}z\end{align*}z must be equal to 4.

To solve a more complex equation, such as \begin{align*}z\times 7 = 105\end{align*}z×7=105, you should use another strategy for solving the equation.

To solve an equation in which a variable is multiplied by a number, you can use the inverse operation of multiplication-division. To isolate the variable you divide both sides of the equation by that number to find the value of the variable.

You can divide both sides of the equation by the same number and not change the equality because of the Division Property of Equality, which states:

If \begin{align*}a=b\end{align*}a=b and \begin{align*}c \neq 0\end{align*}c0, then \begin{align*}\frac{a}{c} = \frac{b}{c}\end{align*}ac=bc.

This means that if you divide one side of an equation by a nonzero number, \begin{align*}c\end{align*}c, you must divide the other side of the equation by that same number, \begin{align*}c\end{align*}c, to keep the values on both sides equal.

Let’s look at an example.

Solve for \begin{align*}z\end{align*}z.

\begin{align*}z \times 7 = 105\end{align*}

z×7=105

In this equation, \begin{align*}z\end{align*}z is multiplied by 7. So, to isolate the variable \begin{align*}z\end{align*}z, you can divide both sides of the equation by 7.

First, using the division property of equality, divide both sides of the equation by 7.

\begin{align*}\begin{array}{rcl} z \times 7 &=& 105 \\ \frac{z \times 7}{7} &=& \frac{105}{7} \end{array}\end{align*}

z×7z×77==1051057

Next, separate the fraction \begin{align*}\frac{z \times 7}{7}\end{align*}z×77, and simplify.

\begin{align*}\begin{array}{rcl} z \times \frac{7}{7} &=& 15 \\ z \times 1 &=& 15 \\ z &=& 15 \end{array}\end{align*}

z×77z×1z===151515

The answer is \begin{align*}z = 15\end{align*}z=15.

Here is another example.

Solve for \begin{align*}r\end{align*}r.

\begin{align*}-8r = 128\end{align*}

8r=128

In this equation, -8 is multiplied by \begin{align*}r\end{align*}r. So, using the division property of equality, you can divide both sides of the equation by -8 to solve for \begin{align*}r\end{align*}r.

First, divide both sides of the equation by -8.

\begin{align*}\begin{array}{rcl} -8r &=& 128 \\ \frac{-8r}{-8} &=& \frac{128}{-8} \end{array}\end{align*}

8r8r8==1281288

Next, separate the fraction \begin{align*}\frac{-8r}{-8}\end{align*}8r8, and simplify.

\begin{align*}\begin{array}{rcl} \frac{-8r}{-8} &=& \frac{128}{-8} \\ 1r &=& -16 \\ r &=& -16 \end{array}\end{align*}

8r81rr===12881616

The answer is \begin{align*}r = -16\end{align*}r=16.

Guided Practice

Sarvenaz earns $8 for each hour she works. She earned a total of $168 last week.

  1. Write an equation to represent \begin{align*}h\end{align*}h, the number of hours she worked last week.
  2. Determine how many hours Sarvenaz worked last week.

First, complete part a.

Let \begin{align*}h\end{align*}h be the number of hours Sarvenaz worked. She earns $8 for each hour she works, so you multiply the number of hours she worked by $8 to find the total amount she earned. Write a multiplication equation.

\begin{align*}8h = 168\end{align*}

8h=168

Next, work on part b.

Solve the equation \begin{align*}8h = 168\end{align*}8h=168 to find \begin{align*}h\end{align*}h, the number of hours she worked last week.

First, use the division property of equality to divide both sides of the equations by 8.

\begin{align*}\begin{array}{rcl} 8h &=& 168 \\ \frac{8h}{8} &=& \frac{168}{8} \end{array}\end{align*}

8h8h8==1681688

Next, separate the fraction \begin{align*}\frac{8h}{8}\end{align*}8h8 and simplify.

\begin{align*}\begin{array}{rcl} \frac{8}{8} h &=& 21 \\ 1h &=& 21 \\ h &=& 21 \end{array}\end{align*}

88h1hh===212121

The answer is Sarvenaz works 21 hours last week.

Examples

Solve each equation.

Example 1

\begin{align*}-4x = 12\end{align*}

4x=12

First, use the division property of equality, and divide both sides of the equation by -4.

\begin{align*}\begin{array}{rcl} -4x &=& 12 \\ \frac{-4x}{-4} &=& \frac{12}{-4} \end{array}\end{align*}

4x4x4==12124

Next, separate the fraction \begin{align*}\frac{-4x}{-4}\end{align*} and simplify.

\begin{align*}\begin{array}{rcl} \frac{-4}{-4} x &=& -3 \\ 1x &=& -3 \\ x &=& -3 \end{array}\end{align*}

The answer is \begin{align*}x = -3\end{align*}.

Example 2

\begin{align*}8a = 64\end{align*}

First, use the division property of equality and divide both sides of the equation by 8.

\begin{align*}\begin{array}{rcl} 8a &=& 64 \\ \frac{8a}{8} &=& \frac{64}{8} \end{array}\end{align*}

Next, separate the fraction \begin{align*}\frac{8a}{8}\end{align*} and simplify.

\begin{align*}\begin{array}{rcl} \frac{8}{8} a &=& 8 \\ 1a &=& 8 \\ a &=& 8 \end{array}\end{align*}

The answer is \begin{align*}a = 8\end{align*}.

Example 3

\begin{align*}9b = 81\end{align*}

First, use the division property of equality and divide both sides of the equation by 9.

\begin{align*}\begin{array}{rcl} 9b &=& 81 \\ \frac{9b}{9} &=& \frac{81}{9} \end{array} \end{align*}

Next, separate the fraction \begin{align*}\frac{9b}{9}\end{align*} and simplify.

\begin{align*}\begin{array}{rcl} \frac{9}{9} b &=& 9 \\ 1b &=& 9 \\ b &=& 9 \end{array}\end{align*}

The answer is \begin{align*}b = 9\end{align*}.

Follow Up

Credit: Dean Hochman
Source: https://www.flickr.com/photos/deanhochman/14757642237/in/photolist-ou5LTx-91d1jK-gSZiNC-fM5Fda-k4ihHF-hbT1br-a9fi3G-qf2NVk-7KzBXp-oEVyQo-8CMehR-6yr59a-eR5B9a-ev6v8m-6yNSWS-bsXcYC-bWXjUw-k4iXhn-nXgM94-qaEMp7-bJrr8T-a7msSe-pX1crj-oRVzJg-pxUarW-bAhHuS-iTdFC5-ABpTx-7btUB6-8n6vSf-4GZHSm-hYWteT-8n6vPj-iDbfpP-f8ecTT-5maKq1-6yaZjW-aqr4XN-ehE6tk-i2EUcw-d9aAvh-gchYqd-qeYrLQ-6xarzP-4Ag76i-8y25gs-a7nKAA-py8EMn-gkatRL-ecZuwP
License: CC BY-NC 3.0

Remember Mr. Ricky’s Biology class?

The class has raised 68 dollars for their trip and is wondering how many 5 dollar tickets they can buy.

First, write an equation to represent this information. Let \begin{align*}d\end{align*}, be the number of tickets they can buy. You can say that, \begin{align*}d\end{align*} times the price of each ticket, 5 dollars, is 68 dollars.

\begin{align*}5d=68\end{align*}

Next, use the division property of equality and divide both sides of the equation by 5.

\begin{align*}\frac{5d}{5}=\frac{68}{5}\end{align*}

Then, separate the fraction \begin{align*}\frac{5d}{5}\end{align*} and simplify.

\begin{align*}\begin{array}{rcl} \frac{5}{5} d &=& 13.6 \\ 1d &=& 13.6 \\ d &=& 13.6 \end{array}\end{align*}

Next, interpret the result.

The students can buy 13.6 tickets. You can’t buy .6 of a ticket. So, the students can only buy 13 tickets with 3 dollars left over.

The answer is the students can buy 13 tickets.

Video Review

https://www.youtube.com/watch?v=zBqIH-E3ero&feature=youtu.be

Explore More

Solve each single variable multiplication equation for the missing value.

  1. \begin{align*}4x = 16\end{align*} 
  2. \begin{align*}6x = 72\end{align*}
  3. \begin{align*}-6x = 72\end{align*}
  4. \begin{align*}-3y = 24\end{align*}
  5. \begin{align*}-3y =-24\end{align*}
  6. \begin{align*}-5x = -45\end{align*}
  7. \begin{align*}-1.4x = 2.8\end{align*}
  8. \begin{align*}3.5a = 7\end{align*}
  9. \begin{align*}7a =-49\end{align*}
  10. \begin{align*}14b = -42\end{align*}
  11. \begin{align*}24b = -48\end{align*}
  12. \begin{align*}-24b = -48\end{align*}
  13. \begin{align*}34b =-102\end{align*}
  14. \begin{align*}84x = 252\end{align*}
  15. \begin{align*}-84x = -252\end{align*}

Vocabulary

Division Property of Equality

Division Property of Equality

The division property of equality states that two equal values remain equal if both are divided by the same number. For example: If 2x = 8, then 1x = 4.
Inverse Operation

Inverse Operation

Inverse operations are operations that "undo" each other. Multiplication is the inverse operation of division. Addition is the inverse operation of subtraction.
Isolate the variable

Isolate the variable

To isolate the variable means to manipulate an equation so that the indicated variable is by itself on one side of the equals sign.

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