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Solving Equations with Distribution

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[MUSIC PLAYING] Let's look at our objectives for today. We'll start by looking at how the distributive property is used when solving equations. We'll then do some examples using the distributive property with a negative number. We'll review how to solve equations. And finally, we'll do some examples using the distributive property in solving equations.

Let's start by looking at how the distributive property is used in solving equations. The distributive property is often used in order to simplify an equation before solving. The distributive property says a times b plus c is equal to a times b plus a times c. This means that we multiply a by each term inside the parentheses. In other words, the outside factor is distributed into the inside factor, which is why it's called the "distributive property."

Let's look at an example. We have 3 times 2 plus 5. Using the distributive property gives us 3 times 2 plus 3 times 5.

We simplify this, starting with the multiplication. 3 times 2 is 6, and 3 times 5 is 15. So we have 6 plus 15, which is 21.

We can see that the distributive property is equivalent to simplifying inside the parentheses first and then multiplying by 3. If we add 2 plus 5 in the parentheses, that gives us 7. And multiplying 3 times 7 gives us 21. So we can see that, in using both methods, it gives us the same value. The distributive property has applications in everyday life, including computing total cast, grade point average, and applying taxes to goods and services.

Now, let's look at how we distribute a negative number. For example, if we have negative 4 times 1 plus 6, this is going to be equal to negative 4 times 1 plus negative 4 times 6. Simplifying, starting with multiplication, gives us negative 4 plus negative 24. So notice that, after distributing, the signs of both numbers in the parentheses have changed here from positive to negative. Negative 4 plus negative 24 gives us a final answer of negative 28.

Here's a second example. We have negative 10 plus 5. When given an expression with only a negative sign outside of the parentheses, we simplify it by distributing a negative 1 into the parentheses. So we have negative 1 times 10 plus negative 1 times 5. Multiplying gives us negative 10 plus negative 5, which is negative 15.

Now, let's briefly review how to solve an equation. In order to solve an equation for a variable, we need to isolate the variable on one side of the equation. To do this, we use inverse operations to isolate the variable. And it's also necessary to combine any like terms on either side of the equation before using inverse operations to solve.

Let's look at an example of solving an equation using the distributive property. Suppose in an isosceles triangle the two congruent angles measure to be 30 degrees more than 7 times the third angle, as shown in the picture below. What is the measure of the third angle?

In this problem, we want to find the measure of the third angle. So we'll define x. A variable x is the measure of the third angle.

We also know that the two other congruent angles are 30 degrees more than 7 times our variable x, which is the third angle. So we're going to multiply 2-- because there's two angles-- by 7x plus 30. The two congruent angles are 30 degrees more than 7 times x. We then add x, the measure of our third angle. And we know that this expression is going to be equal to 180 because all angles in every triangle add up to 180 degrees.

So we start to solve this equation by using our distributive property and multiply 2 times 7x and 2 times 30. This gives us 14x plus 60. We bring down the plus x at the end, which equals 180. We then combine like terms by adding 14x plus x or 1x, which gives us 15x. So our equation is 15x plus 60 equals 180.

We then subtract 60 from both sides of the equation, which gives us 15x equals 120. And then we divide by 15 on both sides, which gives us x equals 8. The measure of the third angle is 8 degrees. We can verify this solution by substituting 8 back into x into our original equation. This gives us 2 times 7 times 8 plus 30 plus 8 equals 180.

We start by simplifying inside our parentheses by multiplying 7 times 8, which is 56. So we have 2 times 56 plus 30 plus 8 equals 180. We then add within our parentheses. 56 plus 30 is 86, so we have 2 times 86 plus 8 equals 180. We then multiply 2 times 86 as 172, so we have 172 plus 8 equals 180.

And finally, adding gives us 180 equals 180, which is a true statement. So our answer is correct.

Let's look at another example of solving an equation using the distributive property. Suppose a school club is buying t-shirts for a school fundraiser. The first 10 t-shirts cost $8 each, but the remaining t-shirts are $4 off the original price. How many shirts can they buy if they have $500 to spend?

So we can let x equal the number of shirts that they can buy. To write our equation, we start by multiplying x by 8, which is the price per t-shirt. They will save $4 for the t-shirts they buy after the first 10 t-shirts, so we subtract 4 times x minus 10. And this will equal the total amount they have to spend, which is $500.

To solve this equation, we start by distributing the negative 4 in front of the parentheses, x minus 10. Negative 4 times x is negative 4x, and negative 4 times negative 10 is a positive 40. So our equation becomes 8x minus 4x plus 40 equals 500.

We then combine like terms, our 8x and the minus 4x. This gives us 4x. So our equation is 4x plus 40 equals 500. We then subtract 40 on both sides, which gives us 4x equals 460. And finally, we would divide by 4 on both sides, which gives us x equals 115. So they can buy 115 t-shirts.

We can verify that the solution is correct by substituting 115 in for x in our original equation. Doing so gives us 8 times 115 minus 4 times 115 minus 10 equals 500. We start by simplifying in our parentheses. 115 minus 10 is 105. So we have 8 times 115 minus 4 times 105 equals 500.

We then multiply 8 times 115, which is 920, and negative 4 times 105, which is negative 420, which equals 500. We then subtract 920 minus 420, which equals 500. So we can see that our solution is correct.

Let's review our important points from today. When given an expression with only a negative sign outside of the parentheses, simplify by distributing a negative 1 to the terms inside the parentheses. And when solving an equation, it is necessary to use the distributive property and combine any like terms on either side of the equation before using the inverse operations to solve.

So I hope that these important points and examples helped you understand a little bit more about solving equations with distribution. Keep using your notes, and keep on practicing. And soon, you'll be a pro. Thanks for watching.

Formulas to Know

- Distributive Property