6.04 One-step equations with multiplication and division

Just like we saw with addition and subtraction, multiplication and division are also inverse operations. For example, multiplying a number by two is the opposite of dividing it by two.

Looking at a balance scale model again, we can see how we can multiply or divide by the same number on both sides of an equation to keep it balanced.

Multiplication property of equality: Multiplying each side of an equation by the same number produces an equivalent equation. Example:

\begin{aligned}&\text{If } &x &= 3 \\ &\text{Then } &x\cdot{2} &= 3\cdot{2}\end{aligned}

We can visualize this with a scale.

The scale tells us that x=3. We can double the values on both sides and the scale will still be balanced.

We multiplied the tiles on both sides of the scale by 2 and it is still balanced. We could have also multiplied both sides of the original scale by 3 to create 3 equal groups on each side of the balance and it would still be balanced.

If we treat each side of the scale like a group that aligns with the other side, we can keep applying the multiplication property of equality.

Division property of equality: Dividing each side of an equation by the same number produces an equivalent equation. Example:

\begin{aligned}&\text{If } &4x &= 8 \\ &\text{Then } &\dfrac{4x}{4} &= \dfrac{8}{4}\end{aligned}

We can visualize this with a scale.

Notice that the groups on the left and right align where every 2x aligns with 4 unit tiles. We can divide both sides by 2 (removing half of the tiles) and the scale will still be balanced.

We can also divide both sides of the original scale by 4, since we can see there are 4 groupings where every x tile is equal to 2 unit tiles.

By dividing both sides of the scale by the same amount, we ensure that the left and right side of the scale are still the same size and weight to keep the scale balanced.

When we use these properties with specific numbers to eliminate them on one side of the equation, we are applying the multiplicative inverse and identity properties.

For example, with the equation 8x=30 we want to isolate x which requires getting rid of the coefficient 8. To do this we can use the inverse operation of dividing by 8.

\dfrac{8x}{8}=\dfrac{30}{8}

This can also be written as multiplication: \dfrac{1}{8} \cdot 8x=\dfrac{1}{8}\cdot 30

This allows us to see the multiplicative inverse in action: 1\cdot x=\dfrac{1}{8}\cdot 30

Now applying the multiplicative identity: x=\dfrac{1}{8}\cdot 30

Then we can simplify the right side of the equation: x=\dfrac{30}{8}=\dfrac{15}{4}

We don't always write out all of these steps, but it is still important to know what is happening algebraically.

Examples

Example 1

Scale 1 is balanced.

Scale 1:

Scale 2:

Which of the following options could go in place of the question mark to balance scale 2?

A

B

C

D

Worked Solution

Create a strategy

From Scale 1, we can say that 1 positive x tile is equivalent to 3 positive 1 tiles or simply x=3.

Apply the idea

We need 9 positive 1 tiles in place of the question mark to balance scale 2, option D.

Reflect and check

We can write this algebraically as:

\displaystyle x	\displaystyle =	\displaystyle 3	Write the equation
\displaystyle 3x	\displaystyle =	\displaystyle 3 \cdot 3	Multiply both sides by 3
\displaystyle 3x	\displaystyle =	\displaystyle 9	Evaluate the multiplication

Watch question walkthrough

Example 2

Solve 3x=18

Worked Solution

Create a strategy

To undo multiplication, we can divide both sides of the equation.

Apply the idea

\displaystyle 3x	\displaystyle =	\displaystyle 18	Write the original equation
\displaystyle \dfrac{3x}{3}	\displaystyle =	\displaystyle \dfrac{18}{3}	Divide both sides by 3
\displaystyle x	\displaystyle =	\displaystyle 6	Evaluate the division

Watch question walkthrough

Example 3

Solve: \dfrac{x}{8}=6

Worked Solution

Create a strategy

To undo division, we can multiply both sides of the equation.

Apply the idea

\displaystyle \dfrac{x}{8}	\displaystyle =	\displaystyle 6	Write the original equation
\displaystyle \dfrac{x}{8}\cdot8	\displaystyle =	\displaystyle 6\cdot8	Multiply both sides by 8
\displaystyle x	\displaystyle =	\displaystyle 48	Evaluate the multiplication

Watch question walkthrough

Example 4

5 is a solution to the equation 8x=40.

a

Verify using substitution.

Worked Solution

Create a strategy

Substitute 5 into the equation. The left side of the equation should be equal to the right side if 5 is a solution.

Apply the idea

\displaystyle 8\left(5\right)	\displaystyle =	\displaystyle 40	Substitute x=5
\displaystyle 40	\displaystyle =	\displaystyle 40	Evaluate the multiplication

The left side of the equation is equal to the right side of the equation, verifying that 5 is a solution.

Watch question walkthrough

b

Verify using a model.

Worked Solution

Create a strategy

Represent the equation using algebra tiles on a scale. Make the left side of the scale to have the same number of unit tiles as the right side by replacing the variable tile with the appropriate tiles.

Apply the idea

The equation 8x=40 can be represented by:

The given solution if 5, so we replace each variable tile with 5 unit tiles. Notice that the variable tile is positive, so the resulting 5 unit tiles are positive.

Both sides of the scale now have 40 positive unit tiles, verifying that 5 is a solution to the equation.

Watch question walkthrough

Example 5

At a beach fruit stand, fresh squeezed juice is sold at \$ 6 per quart. You and your friends spent a total of \$ 84 on quarts of juice. Write an equation that shows the relationship between the total cost and the number of quarts of juice purchased.

Worked Solution

Create a strategy

The number of quarts of juice purchased is the unknown value.

Apply the idea

Let q represent the number of quarts of juice purchased.

6q=84

Watch question walkthrough

Example 6

Write a situation that could represent the equation \dfrac{x}{4}=52.

Worked Solution

Create a strategy

Think of a situation where there is an unknown value, x, divided into 4 equal sized groups and each group is size 52.

Apply the idea

At a craft store, you buy 4 packs of markers. Each pack contains the same number of markers. In total, you have 52 markers. Write an equation that shows the relationship between the number of markers in a pack and the total number of markers you bought.

Watch question walkthrough