**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 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

**Solve for the variable:**

- \(7{\text{L}} = 14\) (Solution
- \(5{\text{Z}} = 20\) (Solution
- \(5{\text{H}} = 25\) (Solution
- \(4{\text{U}} = -24\) (Solution
- \(7{\text{W}} = 63\) (Video Solution
- \(-2{\text{b}} = 16\) (Video Solution