One-step multiplication & division equations

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:

$7 \cdot 3 \div 3 = 7$

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:

$8 \div 4 \cdot 4 = 8$

##Solving a multiplication equation using inverse operations

Let's think about how we can solve for $t$ in the following equation:

$\qquad 6t = 54$

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:

$\begin{align} 6t &= 54 \\
\dfrac{6t}{\blueD{6}} &= \dfrac{54}{\blueD{ 6}}~~~~~~~~~~\small\gray{\text{Divide each side by six.}} \\
t &= \greenD{9}~~~~~~~~~~\small\gray{\text{Simplify.}} \end{align}$

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:

$\qquad$ $\begin{align} 6t &= 54 \
6 \cdot \greenD9 &\stackrel{\large?}{=} 54\
54 &= 54 \end{align}$

Yes, $t = \greenD{9}$ is a solution!

##Solving a division equation using inverse operations

Now, let's try to solve a slightly different type of equation:

$\qquad \dfrac x5 = 7$

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:

$\begin{align} \dfrac x5 &= 7 \\
\dfrac x5 \cdot \blueD{5} &= 7 \cdot \blueD{5}~~~~~~~~~~\small\gray{\text{Multiply each side by five.}} \\
x &= \greenD{35}~~~~~~~~~~\small\gray{\text{Simplify.}} \end{align}$

Let's check our work.

$\qquad$ $\begin{align} \dfrac x5 &= 7 \\
\dfrac{\greenD{35}}{5} &\stackrel{\large?}{=} 7\\
7 &= 7 \end{align}$

Yes, $x = \greenD{35}$ 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:

Let's try solving equations.

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