This tutorial covers solving equations with distribution, through the definition and discussion of:
 The Distributive Property
 The Distributive Property and Negative Numbers
 Solving an Equation Using the Distributive Property
1. The Distributive Property
The distributive property is often used to simplify an equation before solving. It states that a times b plus c is equal to a times b plus a times c. This means that you multiply a by each term inside the parentheses. In other words, the outside factor is “distributed” into the inside factor, which is why it is called the distributive property.


Using the distributive property simplifies the expression, as shown below. Note that the distributive property is equivalent to simplifying inside the parentheses first and then multiplying by 3. If you add 2 plus 5 in the parentheses, which equals 7, then multiply by 3, you arrive at the same answer: 21. Therefore, both methods provide the same value.
The distributive property has applications in everyday life, including computing total cost, grade point average, and applying taxes to goods and services.
2. The Distributive Property and Negative Numbers
There are several important things to note when distributing a negative number.

In the following examples, you will notice that after distributing, the signs of both numbers in the parentheses have changed from positive to negative.

As in the second example above, when given an expression with only a negative sign outside of the parentheses, you simplify it by distributing a negative 1 into the parentheses.
3. Solving an Equation Using the Distributive Property
You may recall that to solve an equation for a variable, you need to isolate the variable on one side of the equation using inverse operations. It’s also necessary to combine any like terms on either side of the equation before using inverse operations to solve. At times, you may also need to use the distributive property in solving an equation.

Suppose you have an isosceles triangle in which the two congruent angles measure 30 degrees more than 7 times the third angle, as shown in the picture below. What is the measure of the third angle?

Step 1: In this problem, the measure of the third angle is the unknown quantity. Therefore, you’ll define the variable x as the measure of the third angle.


Step 2: You also know that the two other congruent angles are 30 degrees more than 7 times the variable x, which is the third angle. Therefore, you’re going to multiply 2—because there are two angles—by 7x plus 30. The two congruent angles are 30 degrees more than 7 times x. You then add x, the measure of your third angle. Finally, you know that this expression is going to be equal to 180, because all angles in a triangle add up to 180 degrees.


Step 3: Now you can start to solve this equation by using the distributive property. Multiply 2 times 7x and 2 times 30, which simplifies to the expression 14x plus 60. Bring down the plus x at the end to complete the expression, which equals 180. Combine the like terms by adding 14x plus x (or 1x), which equals 15x. Now your equation is:


Step 4: Next, subtract 60 from both sides of the equation, which simplifies to 15x equals 120. Divide by 15 on both sides to get the solution of x equals 8. The measure of the third angle is 8 degrees.

You can verify this solution by substituting 8 back into your original equation as x. Simplify using the order of operations, which will provide the true statement 180 equals 180, so your answer is correct.

IN CONTEXT
Suppose a school club is buying tshirts for a school fundraiser. The first 10 tshirts cost $8 each, but the remaining tshirts are $4 off the original price. How many shirts can they buy if they have $500 to spend?
Let x equal the number of shirts that they can buy. To write your equation, you start by multiplying x by 8, which is the price per tshirt. They will save $4 for the tshirts they buy after the first 10 tshirts, so you subtract 4 multiplied by (x10). This will equal the total amount they have to spend, which is $500.
To solve this equation, 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. Therefore, your equation becomes:
Next, you combine the like terms, 8x and minus 4x, which equals 4x. Subtract 40 on both sides, simplifying the expression. Finally, divide by 4 on both sides, which provides the solution x equals 115. Therefore, they can buy 115 tshirts.
Remember to verify that your solution is correct by substituting 115 in for x in your original equation. Using the order of operations, start by simplifying in your parentheses, then move on to multiplication and subtraction. Your final equation is 500 equals 500, so you can see that your solution is correct.
Today you learned that 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. You also learned that when given an expression with only a negative sign outside of the parentheses, you simplify by distributing a negative 1 to the terms inside the parentheses.