Solving for a Variable:

Solving for a Variable on One Side Using Multiplication

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

This video uses the word isolate.

Video Source (05:08 mins) | Transcript

As shown in the video, to isolate a variable when it’s being multiplied, we multiply both sides of the equation with the multiplicative inverse of the number. Remember, the multiplicative inverse is the opposite fraction (ex: $3$ and $\frac{1}{3}$ ).

Additional Resources

Khan Academy: Inverse Property of Multiplication (03:16 mins, Transcript)
Khan Academy: One-step Division Equations (11:05 mins, Transcript)
Khan Academy: One-step Multiplication Equations (02:22 mins, Transcript)

Practice Problems

Solve for the variable:

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

$5{\text{Z}} = 20$

$5{\text{H}} = 25$

$4{\text{U}} = -24$

$7{\text{W}} = 63$

$-2{\text{b}} = 16$

Solutions

$2$
$4$
$5$ (Written Solution)

In this example, we want to get the variable ${\text{H}}$ alone on one side of the equal sign in order to find out what it is equal to. ${\text{H}}$ is currently being multiplied by $5$. We 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 $1$.

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

$5 \times {\color{Red} ?} = 1$

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

We multiply both sides of the equation by $\frac{1}{5}$.

Since $\frac{1}{5}$ multiplied to $5$ equals $1$, we are left with $1{\text{H}}$ on the left side.

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

$1{\text{H}}$ is the same as just ${\text{H}}$ since anything times $1$ is itself.

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

Then we multiply across the numerator and denominator when multiplying fractions.

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

$\frac{25}{5}=5$

${\text{H}}=\frac{{\color{Cyan} 25}}{{\color{Cyan} 5}}={\color{Cyan} 5}$

Our final solution: ${\text{H}} = 5$
$-6$ (Written Solution)

In this example, we want to get the variable ${\text{U}}$ alone on one side of the equal sign in order to find out what it is equal to. ${\text{U}}$ is currently being multiplied by $4$. We 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 $1$.

We are looking for $4({\color{Red} ?}) = 1$

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

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

We multiply both sides of the equation by $\frac{1}{4}$.

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

Since $\frac{1}{4}$ multiplied to $4$ equals $1$, we are left with $1{\text{U}}$ on the left side.

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

$1{\text{U}}$ is the same as just ${\text{U}}$ since anything times $1$ is itself.

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

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

$-\frac{24}{4}=-6$

${\text{U}}=\frac{{\color{Cyan} -24}}{{\color{Cyan} 4}}={\color{Cyan} -6}$

Our final solution: ${\text{U}} = -6$
$9$ (Solution Video | Transcript)
$-8$ (Solution Video | Transcript)