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### Course: Pre-algebra>Unit 7

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:
$7\cdot 3÷3=7$
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:
$8÷4\cdot 4=8$

## Solving a multiplication equation using inverse operations

Let's think about how we can solve for $t$ in the following equation:
$\phantom{\rule{2em}{0ex}}6t=54$
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:

### 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:
$\phantom{\rule{2em}{0ex}}$ $\begin{array}{rl}6t& =54\\ 6\cdot 9& \stackrel{?}{=}54\\ 54& =54\end{array}$
Yes, $t=9$ is a solution!

## Solving a division equation using inverse operations

Now, let's try to solve a slightly different type of equation:
$\phantom{\rule{2em}{0ex}}\frac{x}{5}=7$
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:

### Let's check our work.

$\phantom{\rule{2em}{0ex}}$ $\begin{array}{rl}\frac{x}{5}& =7\\ \\ \frac{35}{5}& \stackrel{?}{=}7\\ \\ 7& =7\end{array}$
Yes, $x=35$ 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 equation$6t=54$Divide each side by six.
Division equation$\frac{x}{5}=7$Multiply each side by five.

## Let's try solving equations.

Equation A
Which operation would help solve for $w$?
$8w=72$
$w=$

## Want to join the conversation?

• 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.
• 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.
• If a letter is close to a number how do you solve the equation?
• I'm assuming your referring to something such as 7x = 14
in this case, you would use 7x to say that 7 of that variable is equal to 14 in which case x = 2. This works because 7x is used to refer to multiplication easily during algebra since if you tried 7 x X = 14, it can get a bit confusing. While you can also use
7 ⋅ X = 14 (⋅ being another form of multiplication symbol) but it's much less common.

TLDR; when a number is by a variable it means to multiply the variable by the number.
• this is too easy.
• Then work up higher grades
• 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.
• yes :)
• It is the same question so will the answer change or stay the same?
• I don't get any of this. ... I tried to understand it but I can't.
;-; Can someone help me please? I am confused how u get a fraction number when ur doing this. Help please :/