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# 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
Which operation would help solve for w?
8, w, equals, 72
w, equals

## Want to join the conversation?

• i dont get this, it says "what you do to one side, you do to the other" so if you have x/5=7 you have to multiply each side by 5, so is it 5x5=25 and 7x5=35? if so what happened to the 25?
• When x/5 is multiplied by 5, the 5's are not multiplied together to give 25. Rather, multiplication by 5 undoes (or cancels out) the division by 5 that occurs in the expression x/5. So there's no 25 here. The final result is x = 35.
• the only one I don't understand is the first one,any tips?
• You divide by 8 on each sides so 72/8 =9
• Is there any way i cant get taught better for inverse operation
• If you are looking for another video or some reference to use for inverse operation, I would recommend using Math Antics or Mathceraptops (YouTube channel).This are the places i personally do to for second opinions. I am not sure if Math antics has a video about inverse operation but Mathceraptops sure do.
• So even if you have a fraction it just means to divide but you should multiply?
• Yes exactly so in the last problem you would cancel out the 4 by multiplying it by 4 and what ever you do on one side of the equal sign you have to do on the other so you would multiply 18 by 4 to get 72.
• About the "what you do to one side, you do to the other".
To understand the logic behind this, first let's imagine a beam balance in front of you, it has two blocks on each side each weighing 200 grams, thus the two sides are equal.

However if you add another 100g block on one side, the balance becomes uneven, as in 200g < 300g . To solve this we add another 100g block to the other side, so the beam balance becomes even again, as in 300g = 300g . That's why what you do on one side you also do to the other, when solving an equation.

TL:DR; We do that to avoid a false equation.