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:
space, 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:
6t=546t6=546          Divide each side by six.t=9          Simplify.\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:
space 6t=5469=?5454=54\begin{aligned} 6t &= 54 \\ 6 \cdot \greenD9 &\stackrel{\large?}{=} 54\\ 54 &= 54 \end{aligned}
Yes, t, equals, start color greenD, 9, end color greenD is a solution!

Solving a division equation using inverse operations

Now, let's try to solve a slightly different type of equation:
space, 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:
x5=7x55=75          Multiply each side by five.x=35          Simplify.\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.

space x5=7355=?77=7\begin{aligned} \dfrac x5 &= 7 \\\\ \dfrac{\greenD{35}}{5} &\stackrel{\large?}{=} 7\\\\ 7 &= 7 \end{aligned}
Yes, x, equals, start color greenD, 35, end color greenD 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: 8, w, equals, 72

Which operation would help solve for w?
Choose 1 answer:
Choose 1 answer:

w, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i

Equation B: w, dot, 5, equals, 55

Which operation would help solve for w?
Choose 1 answer:
Choose 1 answer:

w, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i

Equation C: start fraction, a, divided by, 4, end fraction, equals, 18

Which operation would help solve for a?
Choose 1 answer:
Choose 1 answer:

a, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i

Equation D: 18, equals, start fraction, a, divided by, 4, end fraction

Which operation would help solve for a?
Choose 1 answer:
Choose 1 answer:

a, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i

The equation 18, equals, start fraction, a, divided by, 4, end fraction is the exact same as the equation start fraction, a, divided by, 4, end fraction, equals, 18; it's just written differently.