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Lesson 4: One-step multiplication and division equations

# One-step multiplication & division equations

Learn to solve equations like "4x = 20" or "y/3 = 7".
Based on our understanding of the balance beam model, we know that to keep a true equation, we always have to do the same thing to both sides of an equation.
But how do we know what to do to both sides of the equation?

## Multiplication and division are inverse operations

Here's an example of how division is the inverse operation of multiplication:
If we start with 7, multiply by 3, then divide by 3, we get back to 7:
7, dot, 3, divided by, 3, equals, 7
Here's an example of how multiplication is the inverse operation of division:
If we start with 8, divide by 4, then multiply by 4, we get back to 8:
8, divided by, 4, dot, 4, equals, 8

## Solving a multiplication equation using inverse operations

Let's think about how we can solve for t in the following equation:
6, t, equals, 54
We want to get t by itself on the left hand side of the equation. So, what can we do to undo multiplying by 6?
We should divide by 6 because the inverse operation of multiplication is division!
Here's how dividing by 6 on each side looks:
\begin{aligned} 6t &= 54 \\\\ \dfrac{6t}{\blueD{6}} &= \dfrac{54}{\blueD{ 6}}~~~~~~~~~~\small\gray{\text{Divide each side by six.}} \\\\ t &= \greenD{9}~~~~~~~~~~\small\gray{\text{Simplify.}} \end{aligned}

### Let's check our work.

It's always a good idea to check our solution in the original equation to make sure we didn't make any mistakes:
$\qquad$ \begin{aligned} 6t &= 54 \\ 6 \cdot \greenD9 &\stackrel{\large?}{=} 54\\ 54 &= 54 \end{aligned}
Yes, t, equals, start color #1fab54, 9, end color #1fab54 is a solution!

## Solving a division equation using inverse operations

Now, let's try to solve a slightly different type of equation:
start fraction, x, divided by, 5, end fraction, equals, 7
We want to get x by itself on the left hand side of the equation. So, what can we do to cancel out dividing by 5?
We can multiply by 5 because the inverse operation of division is multiplication!
Here's how multiplying by 5 on each side looks:
\begin{aligned} \dfrac x5 &= 7 \\\\ \dfrac x5 \cdot \blueD{5} &= 7 \cdot \blueD{5}~~~~~~~~~~\small\gray{\text{Multiply each side by five.}} \\\\ x &= \greenD{35}~~~~~~~~~~\small\gray{\text{Simplify.}} \end{aligned}

### Let's check our work.

$\qquad$ \begin{aligned} \dfrac x5 &= 7 \\\\ \dfrac{\greenD{35}}{5} &\stackrel{\large?}{=} 7\\\\ 7 &= 7 \end{aligned}
Yes, x, equals, start color #1fab54, 35, end color #1fab54 is a solution!

## Summary of how to solve multiplication and division equations

Awesome! We just solved a multiplication equation and a division equation. Let's summarize what we did:
Type of equationExampleFirst step
Multiplication equation6, t, equals, 54Divide each side by six.
Division equationstart fraction, x, divided by, 5, end fraction, equals, 7Multiply each side by five.

## Let's try solving equations.

Equation A
• Current
Which operation would help solve for w?
8, w, equals, 72