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Solving for a Variable on One Side Using Multiplication
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Sometimes our variable is being multiplied to a number, in this case, we use the multiplicative inverse (which we learned about in our lessons on fractions) to isolate our variable. In all cases when we’re solving for variables, it is important to remember that anything we do to one side of the equation, we must do to the other.

This video uses the word isolate.

Solving for a Variable on One Side Part 2-Multiplication

As shown in the video, to isolate a variable when it’s being multiplied, we multiply both sides of the equation with the multiplicative inverse of the number. Remember, the multiplicative inverse is the opposite fraction (example: $$3$$ and $${\dfrac {1}{3}}$$ ).

### Practice Problems

Solve for the variable:
1. $$7{\text{L}} = 14$$ (
Solution
Solution:
2
)
2. $$5{\text{Z}} = 20$$ (
Solution
Solution:
4
)
3. $$5{\text{H}} = 25$$ (
Solution
Solution: 5
Details:
In this example, we want to get the variable H alone on one side of the equal sign in order to find out what it is equal to. H is currently being multiplied by 5. We can remove the 5 by multiplying both sides by the multiplicative inverse of 5.

The multiplicative inverse of a number is the number that when multiplied to it, the product is 1.

We are looking for:
$$5 \times {\color{Red} ?} = 1$$

The multiplicative inverse of 5 is $$\dfrac{1}{5}$$, because $$5\left (\dfrac{1}{5} \right )=1$$.

We multiply both sides of the equation by $$\dfrac{1}{5}$$.

Since $$\dfrac{1}{5}$$ multiplied to 5 equals 1, we are left with 1H on the left side.

1H is the same as just H since anything times 1 is itself.

On the right-hand side of the equation, $$\dfrac{1}{5}$$ times 25 is the same as $$\dfrac{1}{5}$$ times $$\dfrac{25}{1}$$, since anything divided by 1 is still itself.

Then we multiply across the numerator and denominator when multiplying fractions.

$${\text{H}}=\dfrac{{\color{blue} 25}}{{\color{blue} 5}}={\color{blue} 5}$$

Our final solution: $${\text{H}} = 5$$
)
4. $$4{\text{U}} = -24$$ (
Solution
Solution: $$-6$$
Details:
In this example, we want to get the variable U alone on one side of the equal sign in order to find out what it is equal to. U is currently being multiplied by 4. We can remove the 4 by multiplying both sides by the multiplicative inverse of 4.

The multiplicative inverse of a number is the number that when multiplied to it, the product is 1.

We are looking for:

$$4({\color{Red} ?}) = 1$$

The multiplicative inverse of 4 is $$\dfrac{1}{4}$$, because $$4\left ( \dfrac{1}{4} \right )=1$$.

We multiply both sides of the equation by $$\dfrac{1}{4}$$.

Since $$\dfrac{1}{4}$$ multiplied to 4 equals 1, we are left with 1U on the left side.

1U is the same as just U since anything times 1 is itself.

On the right-hand side of the equation, $$\dfrac{1}{4}$$ times $$-24$$ is the same as $$-24$$ divided by 4 after multiplying across.

$${\text{U}}=\dfrac{{\color{blue} -24}}{{\color{blue} 4}}={\color{blue} -6}$$

Our solution is: $$U=-6$$
)
5. $$7{\text{W}} = 63$$ (
Video Solution
Solution: 9
Details:

(Video Source | Transcript)
)
6. $$-2{\text{b}} = 16$$ (
Video Solution
Solution: $$-8$$
Details:

(Video Source | Transcript)
)

### Need More Help?

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