Solving Systems of Equations by Substitution
Throughout this tutorial, we have dealt with problems that have one equation and usually one
variable to work with. However, many times in algebra we have to deal with problems which
give rise to sets of equations with several variables. If the particular problem gives us a set of
equations, which have the same variables, then we call the set a system of equations. Now, we
must develop techniques for finding the solutions set for multiple variables that will work for all
equations within the system of equations.
We can solve a system of equations by substitution, by elimination, or graphically. We will look
at the substitution method in this section
Substitution Method
In the substitution method, we start with one equation in the system and solve for one variable in
terms of the other variable. We then substitute the result into the other equation. The following
box describes the procedure more explicitly.
SUBSTITUTION METHOD
1. SOLVE FOR ONE VARIABLE. Choose one equation and solve for one variable in
terms of the other variable.
2. SUBSTITUTE. Substitute the expression you found in Step 1 into the other equation
to get an equation in one variable, and then solve for that variable.
3. BACK-SUBSTITUTE. Substitute the value you found in Step 2 back into the
expression found in Step 1 to solve for the remaining variable.
Example 1: Find all solutions of the system.
27
31
xy
xy
+=
⎧
⎨−=
⎩
3
Solution:
Step 1: We solve for y in the first equation.
72
y
x=−
Math 0303
Student Learning Assistance Center - San Antonio College
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