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Substitution in multi-step linear equations

Substitution in multi-step linear equations

Author: Sophia Tutorial
Description:

Use the Substitution Property of Equality to rewrite an expression for another variable.

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Tutorial
what's covered
  1. Substitution Property of Equality
  2. Substituting Expressions into Equations
  3. Substituting to Solve an Equation

1. Substitution Property of Equality

The Substitution Property of Equality allows us to swap or substitute equivalent quantities in expressions and equations. Let's take a look at a basic example of substitution.

2 x plus 3 comma space space space space space space space space space space space x equals 3
Substitute 3 in for x, because they are equal
2 open parentheses 3 close parentheses plus 3
Evaluate 2(3)
6 plus 3
Add 6 and 3
9
Our Solution


2. Substituting Expressions in Equations

Sometimes, we are given an expression for the variable, rather than a single value. We can still use the Substitution Property of Equality to simplify expressions and solve equations. Most often, this requires distribution after substituting, in order to simplify the equation or expression. This is illustrated in the example below:

P equals 0.5 x plus 12 comma space space space space x equals 3 a minus 2
Substitute 3a-2 in for x, because they are equal
P equals 0.5 open parentheses 3 a minus 2 close parentheses plus 12
Distribute (.05) into (3a-2)
P equals open parentheses 1.5 a minus 1 close parentheses plus 12
Combine like terms -1 and 12
P equals 1.5 a plus 11
Our Solution

hint
If there is a coefficient in front of the variable that is substituted with an expression, it will require that we distribute it into the newly substituted expression in order to simplify.


3. Substituting to Solve an Equation

Let's apply the concept of substitution to solve an equation. Suppose you sell gift bags from a kiosk at a local strip mall. Each gift bag costs $7, and you received $15 in tips for the day. We can represent your profit with the equation: R equals space 7 x plus 15, where R is revenue, and x is the number of gift bags sold. (7 is multiplied by x to represent revenue from sales, and 15 is added to account for the tips.)

You figure that you averaged 8 sales per hour, and at the end of the day, a customer bought 10 of them for a party she is attending. We can represent the number of gift bags sold by the equation x space equals space 8 t space plus space 10, where x is the number of gift bags sold, and t is time in hours. (8 is multiplied by t to represent 8 bags sold each hour, and we add 10 to account for the customer who bought 10 for her party.)

Let's take a look at our equations:

R space equals space 7 x space plus space 15

x space equals space 8 t space plus space 10

How long did it take to generate $253 in revenue? Notice that we can substitute $253 in for R, but we want to solve for t, time. One method would be to solve for x, and then substitute that value in order to solve for t. Another method involves making all necessary algebraic substitutions first, and then solving a simplified equation.

253 equals 7 x plus 15 comma space space space x equals 8 t plus 10
Substitute 8 t plus 10 in for x, because they are equal
253 equals 7 open parentheses 8 t plus 10 close parentheses plus 15
Distribute 7 into 8 t plus 10
253 equals 56 t plus 70 plus 15
Combine like terms 70 and 15
253 equals 56 t plus 85
Subtract 85 from both sides
168 equals 56 t
Divide by 56
3 equals t
Our Solution

This means that $253 was generated after 3 hours of selling gift bags. Not bad!

summary
The substitution property of equality states that we can substitute an expression that is equal to some variable into another equation or expression containing that same variable. After you substitute, you can simplify and/or solve the equation.