Math Review of Solving Systems by Substitution

Overview

One of the ways to solve systems of equations is by graphing the equations.  However, graphing the equations is not always the most accurate method to solve them.  If one variable in a system is represented in terms of the other variable in the system, the systems can be solved by substitution.

Using Substitution

Suppose one of the equations in the system is x + y = 5 and the other equation is x = y +1.  The expression y +1 can be substituted for x, so that y +1 +y =5.  Then, there is just one variable so that 2y +1 =5, 2y +1 -1 = 5-1, or 2y = 4, or y =2.  In order to check, substitute the value of y to solve for x, such that x +2 = 5, or x +2-2 = 5-2, or x = 3.  Check the second equation also, so that 3 =2 +1. That is the way to use substitution to solve a system of equations.

Isolating the Variables

Sometimes, the variables cannot be isolated as easily in a system of equations, but the system of substitution can still be used.  Suppose the equations were x-2y = 8 and 2x +y = 8.  The first equation can be rearranged such that x = 8 +2y.  Using substitution, the second equation then becomes 2(8 +2y) +y =8, or 16 +4y +y =8.  As before, there is only one variable, such that 5y = 8-16 or 5y=-8, or y = -8/5.  Again, check the value of x, so that x – (2)(-8/5) =8, or      x +16/5 =8.  (Notice how the sign changes when two negative values are multiplied.) Then multiply both sides by 5, so that 5x +16 = 40, or 5x =24 or x = 24/5.  To check the first equation, 24/5 – 2[-8/5] equals 24/5 +16/5 = 40/5, or 8.  To check the second equation 2 (24/5) – (8/5) = 48/5 – 8/5) = 40/5 = 8.  

Understanding the Problem and Developing a Plan

Math problems that are written in words can often be translated into systems of equations, then solved by using substitution.  Suppose the statement were “The sum of two numbers is 82.  One number is 12 more than the other.  What is the larger number?”  The first sentence can be represented by the equation x +y = 82.  The second sentence can be represented by the equation x=12 +y.

Problem-Solving:  Solving the Problem and Checking the Answer

To solve the problem, take the system of equations and use substitution, so that 12 +y +y = 82, then 2y = 82-12, or 2y = 70, then y = 70/2, or 35.  Using the second equation to solve for x, 12 +35 = 47, and using the first equation, 47 +35 = 82.

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