Solving for a Variable on One Side Using Multiplication

Introduction

In this lesson, you will learn how to solve for a variable on one side of an equation using multiplication.

This video illustrates the lesson material below. Watching the video is optional.

Solving for a Variable on One Side (Part 2): Multiplication (05:08 mins) | Transcript

Solving for a Variable on One Side Using Multiplication

Sometimes the variable is being multiplied to a number, in this case, use the multiplicative inverse (which you learned about in the lessons on fractions) to isolate the variable. In all cases when solving for variables, it is important to remember that anything you do to one side of the equation, you must do to the other.

Example 1
Solve for $x$: $3x = 12$.

In addition and subtraction, use the additive inverse. Remember, the additive inverse is something like $3+(-3)$. 3 and -3 are the additive inverses of each other because when added together, they equal 0.

With multiplication, use the multiplicative inverse. The multiplicative inverse is the fraction opposite the number that, when multiplied by the number, equals 1. What is the multiplicative inverse of 3? $3\times\frac{1}{3}=1$, so $\frac{1}{3}$ is the multiplicative inverse of 3.

Use this to help isolate x. If you multiply one side of the equation by $\frac{1}{3}$, then you'll need to do the same thing to the other side.

\begin{align*}3x &= 12 &\color{red}\small\text{Solve for x}\\\$3x)\color{red}\mathbf{\frac{1}{3}} &= (12) \color{red}\mathbf{\frac{1}{3}} &\color{red}\small\text{Multiplicative inverse of 3 is \(\frac{1}{3}$}\\\\x&= \frac{12}{3} &\color{red}\small\text{Multiply}\\\\x&= 4 &\color{red}\small\text{Simplify}\\\\\end{align*}

Because $\frac{1}{3}$ and 3 are multiplicative inverses of one another, they will cancel each other out and equal 1, so you are left with just x on this side of the equation.
When you multiply fractions, multiply the numerators together and the denominators together. In this example, this gives $\frac{12}{3}$, which is the same as $\frac{12}{3} = 4$, or $12\div3=4$.

The final answer is $x=4$. Now, substitute 4 for x to see if it is really the solution to the equation. $3\times4=12$, so this is the correct solution for x.

Example 2

\begin{align*}
5x &= 30 &\color{red}\small\text{Solve for x}\\\\
(5x)\color{red}\mathbf{\frac{1}{5}} &= (30) \color{red}\mathbf{\frac{1}{5}} &\color{red}\small\text{Multiplicative inverse of 5 is $\frac{1}{5}$}\\\\
x&= \frac{30}{5} &\color{red}\small\text{Multiply}\\\\
x&= 6 &\color{red}\small\text{Simplify}
\end{align*}

Because $5\cdot 6 = 30$, $x=6$ is correct.

Things to Remember

The multiplicative inverse is the fraction opposite a number that, when multiplied by the number, gives you 1.
Remember to switch the numerator and denominator when determining the inverse of any number. For example, $5= \frac{5}{1}$ so the inverse of 5 is $\frac{1}{5}$.

Practice Problems

$7{\text{L}} = 14$ (
Solution

x

Solution: 2

)
$5{\text{Z}} = 20$ (
Solution

x

Solution: 4

)
$5{\text{H}} = 25$ (
Solution

x

Solution: 5
Details:
In this example, you want to get the variable H alone on one side of the equal sign in order to find out what it is equal to. H is currently being multiplied by 5. You can remove the 5 by multiplying both sides by the multiplicative inverse of 5.

The multiplicative inverse of a number is the number that when multiplied to it, the product is 1.

You are looking for:
$5 \times {\color{Red} ?} = 1$

The multiplicative inverse of 5 is $\dfrac{1}{5}$, because $5\left (\dfrac{1}{5} \right )=1$.

Multiply both sides of the equation by $\dfrac{1}{5}$.

Since $\dfrac{1}{5}$ multiplied to 5 equals 1, you are left with 1H on the left side.

$This is a picture of the equation \frac{1(5)}{5}H=\frac{1}{5}(25). There is a vertical dashed line through the equal sign showing the separation of the two halves of the equation. “Left-hand side” is written above \frac{1(5)}{5}H and “Right-hand side” is written above one-fifth (25). 1h=one-fifth(25) is written below \frac{1(5)}{5}H=one-fifth(25).$

1H is the same as just H since anything times 1 is itself.

On the right-hand side of the equation, $\dfrac{1}{5}$ times 25 is the same as $\dfrac{1}{5}$ times $\dfrac{25}{1}$, since anything divided by 1 is still itself.

Then multiply across the numerator and denominator when multiplying fractions.

$This is a picture of the equation H=\frac{1}{5}(\frac{25}{1}). There is a vertical dashed line through the equal sign showing the separation of the two halves of the equation. “Left-hand side” is written above H and “Right-hand side” is written above \frac{1}{5}(\frac{25}{1}). H=\frac{1(25)}{5(1)} is written below the original equation. The next line is: H=\frac{25}{5}.$

${\text{H}}=\dfrac{{\color{blue} 25}}{{\color{blue} 5}}={\color{blue} 5}$

The final solution is: ${\text{H}} = 5$.

)
$4{\text{U}} = -24$ (
Solution

x

Solution: $-6$
Details:
In this example, you want to get the variable U alone on one side of the equal sign in order to find out what it is equal to. U is currently being multiplied by 4. You can remove the 4 by multiplying both sides by the multiplicative inverse of 4.

The multiplicative inverse of a number is the number that when multiplied to it, the product is 1.

You are looking for:

$4({\color{Red} ?}) = 1$

The multiplicative inverse of 4 is $\dfrac{1}{4}$, because $4\left ( \dfrac{1}{4} \right )=1$.

Multiply both sides of the equation by $\dfrac{1}{4}$.

$This is a picture of the equation 4U=-24. There is a vertical dashed line through the equal sign. “Left-hand side” is written above 4U and “Right-hand side” is written above -24. \frac{1}{4}(4U)=\frac{1}{4}(-24) is written below 4U=-24.$

Since $\dfrac{1}{4}$ multiplied to 4 equals 1, you are left with 1U on the left side.

$This is a picture of the equation \frac{1(4)}{4}U= one-fourth(-24). There is a vertical dashed line through the equal sign. “Left-hand side” is written on the left of the dashed line, with “right-hand side” written on the right. 1U=one-fourth(-24) is written below \frac{1(4)}{4}U = one-fourth(-24).$

1U is the same as just U, since anything times 1 is itself.

On the right-hand side of the equation, $\dfrac{1}{4}$ times $-24$ is the same as $-24$ divided by 4 after multiplying across.

$This is a picture of the equation 1U = one-fourth(-24). There is a vertical dashed line through the equal sign. “Left-hand side” is written on the left of the dashed line, with “right-hand side” written on the right. U = \frac{1}{4}(\frac{-24}{1})=\frac{-24}{4} is written below 1U = one-fourth(-24).$

${\text{U}}=\dfrac{{\color{blue} -24}}{{\color{blue} 4}}={\color{blue} -6}$

The solution is: $U=-6$.

)
$7{\text{W}} = 63$ (
Video Solution

x

Solution: 9
Details:

(Solving for a Variable on One Side Using Multiplication #5 (02:14 mins) | Transcript)

| Transcript)
$-2{\text{b}} = 16$ (
Video Solution

x

Solution: $-8$
Details:

(Solving for a Variable on One Side Using Multiplication #6 (02:47 mins) | Transcript)

| Transcript)

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