Hello everyone. In this math post, I cover solving linear equations with the distributive law. This is suitable for early high school mathematics.

## Topics

- Review Of Solving Linear Equations
- Review Of Distributive Law
- Solving Linear Equations With The Distributive Law
- Practice Problems
- Solutions To Practice Problems

## Review Of Solving Linear Equations

When it comes to solving linear equations we seek to find the value of the unknown (such as `x`

) which makes the equation hold true. For the equation `x - 7 = 8`

, the value of the unknown `x`

is 15.

Consider a more involved example where we have the equation `2x - 7 = 15`

. To find the value of `x`

here. We add 7 to both sides to "remove" the negative seven from the left side. This leaves us with:

Computing the right side of 15 and 7 gives 22. We now have `2x = 22`

. To find the value of just `x`

itself, divide by 2 on both sides. The value of the unknown `x`

is just 11.

## Review Of The Distributive Law

The distributive law from algebra allows for multiplication of a single term monomial with a polynomial. One example would be `5(x + 3)`

. Multiplying the 5 through each term in (x + 3) with the distributive law yields `5x + 15`

.

The above example dealt with a monomial with a binomial. What if the binomial was something else that was longer. Let's take a look at another example.

Using the distributive law on `8(x + 2y + z)`

gives `8x + 16y + 8z`

. The distributive law idea still holds. It is just there are more computations involved.

## Solving Linear Equations With The Distributive Law

There are times when you need to certain linear equations that require the distributive law. Once you have a good grasp on using the distributive law you can easily solve for unknowns. Let's look at a few examples.

**Example One**

Solve for x in the equation `9(x + 2) = 22`

.

As soon as you see something like a number multiplied by a binomial in brackets, think of the distributive law first. (You could divide by 9 both sides and subtract by 2 to solve as an alternate approach.)

Isolate for 9x by subtracting 18 on both sides.

Since we want to solve for a single `x`

divide both sides of the equation by 9. The answer for `x`

is four ninths.

**Example Two**

This second example is a bit more involved. Solve for `x`

in the equation 3(x - 7) = 5 + 8(x + 3).

Use distributive law on both sides accordingly. The equation then becomes:

Add the 2 and 24 from the right side.

Collect the `x`

terms in one side and the numbers to the other side.

Simplify the like terms.

To solve for `x`

, divide both sides by negative 5.

## Practice Problems

In each question solve for the value of `x`

.

3(x - 2) = 12

-5(x - 1) = 2

10x - 2 = 3(x - 4)

7(x - 2) = 9(2x + 1)

2(x + y) = 3(x - y)

## Solutions To Practice Problems

6

3/5

-10/7

-23/11

5y