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Substitution in multi-step linear equations
Solving Linear Equations and Inequalities
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# Substitution in multi-step linear equations

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Author: Sophia Tutorial
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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.

 Substitute 3 in for x, because they are equal Evaluate 2(3) Add 6 and 3 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:

 Substitute 3a-2 in for x, because they are equal Distribute (.05) into (3a-2) Combine like terms -1 and 12 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: , 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 , 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:

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.

 Substitute in for x, because they are equal Distribute 7 into Combine like terms 70 and 15 Subtract 85 from both sides Divide by 56 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.

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