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2 Tutorials that teach Solving Equations with Distribution
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Solving Equations with Distribution

Solving Equations with Distribution

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In this lesson, students to learn how to solve equations that involve the distributive property.

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Tutorial
This tutorial covers solving equations with distribution, through the definition and discussion of:
  1. The Distributive Property
  2. The Distributive Property and Negative Numbers
  3. 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.

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

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.
table attributes columnalign left end attributes row cell 3 left parenthesis 2 plus 5 right parenthesis equals 3 cross times 2 plus 3 cross times 5 equals 6 plus 15 equals 21 end cell row cell 3 left parenthesis 2 plus 5 right parenthesis equals 3 left parenthesis 7 right parenthesis equals 21 end cell end table
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.
table attributes columnalign left end attributes row cell negative 4 left parenthesis 1 plus 6 right parenthesis equals negative 4 cross times 1 plus negative 4 cross times 6 equals negative 4 plus negative 24 equals negative 28 end cell row cell negative left parenthesis 10 plus 5 right parenthesis equals negative 1 cross times 10 plus negative 1 cross times 5 equals negative 10 plus negative 5 equals negative 15 end cell end table

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?

File:768-dist1.PNG

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.

x equals m e a s u r e space o f space t h e space 3 r d space a n g l e
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.

2 left parenthesis 7 x plus 30 right parenthesis plus x equals 180
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:

table attributes columnalign left end attributes row cell 2 cross times 7 x plus 2 cross times 30 plus x equals 180 end cell row cell 14 x plus 60 plus x equals 180 end cell row cell 15 x plus 60 equals 180 end cell end table
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.

table attributes columnalign left end attributes row cell 15 x plus 60 minus 60 equals 180 minus 60 end cell row cell 15 x equals 120 end cell row cell fraction numerator 15 x over denominator 15 end fraction equals 120 over 15 end cell row cell space space space space space space space x equals 8 end cell end table
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 left parenthesis 7 left parenthesis 8 right parenthesis plus 30 right parenthesis plus 8 equals 180 end cell row cell 2 left parenthesis 56 plus 30 right parenthesis plus 8 equals 180 end cell row cell 2 left parenthesis 86 right parenthesis plus 8 equals 180 end cell row cell 172 plus 8 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.

table attributes columnalign left end attributes row cell x equals n u m b e r space o f space t minus s h i r t s space p u r c h a s e d end cell row cell 8 x minus 4 left parenthesis x minus 10 right parenthesis equals 500 end cell end table

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:

8 x minus 4 x plus 40 equals 500

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 t-shirts.

table attributes columnalign left end attributes row cell 4 x plus 40 equals 500 end cell row cell 4 x plus 40 minus 40 equals 500 minus 40 end cell row cell 4 x equals 460 end cell row cell fraction numerator 4 x over denominator 4 end fraction equals 460 over 4 end cell row cell space space space x equals 115 end cell end table

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 left parenthesis 115 right parenthesis minus 4 left parenthesis 115 minus 10 right parenthesis equals 500 end cell row cell 8 left parenthesis 115 right parenthesis minus 4 left parenthesis 105 right parenthesis equals 500 end cell row cell 920 minus 420 equals 500 end cell row cell 500 equals 500 end cell end table

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.