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Solving Equations with Distribution

Author: Sophia

what's covered
This tutorial covers solving equations with distribution, through the definition and discussion of:

Table of Contents

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.

formula to know
Distributive Property
a left parenthesis b plus c right parenthesis equals a b plus a c

EXAMPLE

Use the distributive property to simplify the expression: 3 open parentheses 2 plus 5 close parentheses.

3 open parentheses 2 plus 5 close parentheses Our Expression
open parentheses 3 cross times 2 close parentheses plus open parentheses 3 cross times 5 close parentheses We distribute the three and multiply it by both numbers in the parentheses
6 plus 15 3 times 2 is 6 and 3 times 5 is 15.
21 Our Solution

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.

3 open parentheses 2 plus 5 close parentheses Our Expression
3 stack open parentheses 2 plus 5 close parentheses with underbrace below First, we can add the two numbers in the parentheses. 2 plus 5 equals 7.
3 open parentheses 7 close parentheses Three times seven equals 21
21 Our Solution

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. When you distribute a negative number, the signs of the numbers in the parentheses will switch to the opposite sign.

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

EXAMPLE

Evaluate short dash 4 open parentheses 1 plus 6 close parentheses.

short dash 4 open parentheses 1 plus 6 close parentheses Our Expression
open parentheses short dash 4 cross times 1 close parentheses plus open parentheses short dash 4 cross times 6 close parentheses Distribute negative 4 into the parentheses
short dash 4 plus short dash 24 Evaluate
short dash 28 Our Solution

EXAMPLE

Evaluate short dash open parentheses 10 plus 5 close parentheses.

short dash open parentheses 10 plus 5 close parentheses Our Expression
open parentheses short dash 1 cross times 10 close parentheses plus open parentheses short dash 1 cross times 5 close parentheses Distribute negative 1 into the parentheses
short dash 10 plus short dash 5 Evaluate
short dash 15 Our Solution


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.

EXAMPLE

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



In the problem, the measure of the third angle is the unknown quantity. We will define the variable x as the measure of the third angle.

2 open parentheses 7 x plus 30 close parentheses plus x equals 180 We also know that the other two congruent angles are each 30 degrees more than 7 times the variable x. This means we must multiply 2 (because there are two angles) by 7x plus 30. You then add x, the measure of your third angle. Finally, you know that this expression is going to equal 180 because all angles in a triangle add up to 180 degrees.
open parentheses 2 cross times 7 x close parentheses plus open parentheses 2 cross times 30 close parentheses plus x equals 180 Next, we will start to solve the equation by using the distributive property. Multiply 2 times 7x and 2 times 30. This will simplify to 14x plus 60. Include the x at the end to complete the expression.
14 x plus 60 plus x equals 180 Combine the like terms x and 14x to get 15x.
table attributes columnalign left end attributes row cell 15 x plus 60 equals 180 end cell row cell space space space space space space space minus 60 space space space minus 60 end cell end table Subtract 60 from both sides to isolate the x variable.
table attributes columnalign left end attributes row cell 15 x equals 120 end cell row cell space stack 15 space with bar on top space space space space space space 15 with bar on top end cell end table Divide both sides by 15 to get the solution.
x equals 8 Our Solution

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.

table attributes columnalign left end attributes row cell 2 open parentheses stack 7 open parentheses 8 close parentheses with underbrace below plus 30 close parentheses plus 8 equals 180 end cell row cell 2 open parentheses stack 56 plus 30 with underbrace below close parentheses plus 8 equals 180 end cell row cell stack 2 open parentheses 86 close parentheses with underbrace below plus 8 equals 180 end cell row cell stack 172 plus 8 with underbrace below equals 180 end cell row cell 180 equals 180 end cell end table

IN CONTEXT

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?

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 t-shirt. They will save $4 for the t-shirts they buy after the first 10 t-shirts, so you subtract 4 multiplied by (x-10). This will equal the total amount they have to spend, which is $500.

Let's use x for the number of t-shirts purchased.

8 x minus 4 open parentheses x minus 10 close parentheses equals 500 Our Equation
8 x plus short dash 4 open parentheses x minus 10 close parentheses equals 500 On the left side, 8 x minus 4 open parentheses x minus 10 close parentheses can be written as 8 x plus short dash 4 open parentheses x minus 10 close parentheses and now we can distribute the -4.
8 x plus open parentheses open parentheses short dash 4 cross times x close parentheses minus open parentheses short dash 4 cross times 10 close parentheses close parentheses equals 500 Start by distributing the -4 in front of the parentheses.
8 x plus left parenthesis short dash 4 x minus short dash 40 right parenthesis equals 500 -4 times x is -4x. -4 times 10 is -40.
8 x plus open parentheses short dash 4 x plus 40 close parentheses equals 500 In the parentheses, short dash 4 x minus short dash 40 can be rewritten as short dash 4 x plus 40.
8 x minus 4 x plus 40 equals 500 Once we remove the parentheses, combine like terms. 8x minus 4x equals 4x.
table attributes columnalign left end attributes row cell 4 x plus 40 equals 500 end cell row cell space space space space minus 40 space space space minus 40 end cell end table Subtract 40 from both sides to isolate the x variable.
table attributes columnalign left end attributes row cell 4 x equals 460 end cell row cell stack space 4 space with bar on top space space space space space space stack space 4 space with bar on top end cell end table Divide 4 from each side.
x equals 115 Our Solution

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.

table attributes columnalign left end attributes row cell 8 open parentheses 115 close parentheses minus 4 open parentheses stack 115 minus 10 with underbrace below close parentheses equals 500 end cell row cell stack 8 open parentheses 115 close parentheses with underbrace below minus stack 4 open parentheses 105 close parentheses with underbrace below equals 500 end cell row cell stack 920 minus 420 with underbrace below equals 500 end cell row cell 500 equals 500 end cell row blank row blank end table

summary
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.

Source: This work is adapted from Sophia author Colleen Atakpu.

Formulas to Know
Distributive Property

a left parenthesis b plus c right parenthesis equals a b plus a c