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Substitution in multi-step linear equations

Substitution in multi-step linear equations

Author: Sophia Tutorial

This lesson shows how to substitute expressions for variables in equations.

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What's Covered

  • Substitution
  • Substituting Expressions into Equations
  • Solving Equations using Substitution 

Substitution in Multi-Step Linear Equations


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.


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:


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.  


Substituting an Expression 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. 


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