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SO WHAT IS DISTRIBUTION?

So we’ve gone over the distributive property before, but let’s officially define it.

Again, according to Google, the distributive property is:

“An algebra property which is used to multiply a single term and two or more terms inside a set of parentheses.”

So in other words, we can distribute things if we are multiplying a single number of variable by an equation.

Seems sort of complicated, but we’ve been doing it for a while. But, just in case, here’s an example to clarify:

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Example 1:

Let’s say we are given:

-(x+5) = 7

Well, we know there is a negative outside of the parenthesis, so we need to distribute the negative.

We do that by multiplying each term by the negative.

So:

-(x+5) = 7

So:

-1 * x = -x

And

-1 * 5 = -5

So now we are left with:

-x – 5 = 7

+ 5 + 5

-x = 12

x = -12

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REVIEW OF DISTRIBUTION:

Let’s say we are given:

-(x-6) = 13

Well, we know there is a negative outside of the parenthesis, so we need to distribute the negative.

We do that by multiplying each term by the negative.

So:

-(x - 6) = 13

So:

-1 * x = -x

And

-1 * -6 = +6 (or just 6)

So now we are left with:

-x + 6 = 13

- 6 - 6

-x = 7

x = -7

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THAT’S PRETTY MUCH IT

The distributive property is just that, distributing the terms.

So why is this useful?

Well, it’s useful because it helps us find a term we may be missing, and we may not be able to find otherwise.

A good example of this would be:

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Example 2

3(x + 5) = 45

Again, we have no idea what x is, but we do know that 3 is being multiplied by x.

So let’s distribute that:

3(x + 5) = 45

So:

3 * x = 3x

3 * 5 = 15

Now we have:

3x + 15 = 45

- 15 - 15

3x = 30

3 3

x = 10

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Example #3:

-3(2x - 15) = 75

Again, we know that -3 is being multiplied by the equation.

So let’s distribute that -3:

-3(2x - 15) = 75

So:

-3 * 2x = -3 * 2 * x = -6 * x = -6x

-3 * -15 = 45

Now we have:

-6x + 45 = 75

- 45 - 45

-6x = 30

-6 -6

x = -5

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BUT HOW DO WE KNOW THIS WORKS?

We know this works because when we use numbers instead of variables, it works.

So:

4(7 – 5)

We know that 7 – 5 = 2, so:

4(2) = 8

Now let’s distribute! If it works, then we should get an answer of 8 still.

So:

4(7 – 5)

4 * 7 = 28

4 * 5 = 20

28 – 20 = 8

It worked!

And that’s how we know the distributive property works.

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So now we get to the next level

Again, simply multiplying a number by an equation is the easiest way to multiply it out.

We know it’s much harder to multiply an equation by a variable instead of a number.

So, let’s review really quick over what happens when we are presented an equation being multiplied by a variable.

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Example #4:

-3(4x - 12) = 72

Again, we know that -3 is being multiplied by the equation.

So let’s distribute that -3:

-3(4x - 12) = 72

So:

-3 * 4x = -3* 4 * x = -12 * x = -12x

-3 * -12 = 36

Now we have:

-12x + 36 = 72

- 36 - 36

-12x = 36

-12 -12

x = -3

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Example #5:

5(4x + 10) = 100

Again, we know that 5 is being multiplied by the equation.

So let’s distribute that 5:

5(4x + 10) = 100

So:

5 * 4x = 5 * 4 * x = 20 * x = 20x

5 * 10 = 50

Now we have:

20x + 50 = 100

- 50 - 50

20x = 50

20 20

x = 5/2