Solving Systems of Equations using Substitution
To solve a system of equations without graphing, you can use
the substitution method shown in the Example
below. In general, if you solve a system of equations and the
result is a true statement, such as - 5 = -5, the system has
infinitely many solutions; if the result is a false statement,
such as - 5 = 7, the system has no solution .
Use substitution to solve the system of equations x + y = 1
and 2x + y = - 1.
Step 1: Solve one of the equations for x or
x + y = 1 Solve the first equation for x since the coefficient
of x is 1.
x = 1- y
Step 2: Substitute this value into the other
|2x + y = - 1
||Use the second equation.
|2(1 - y) + y = -1
||Substitute 1 - y for x.
|2 - 2y + y = -1
Step 3: Solve this equation.
2 - 2y + y = -1 Solve for y.
-y = -3 or y = 3
Step 4: Find the value of the other variable
using substitution into either equation.
|x + y = 1
||Use the first equation.
|x + 3 = 1
||Substitute 3 for y.
|x = -2
||Solve for x.
The solution to the system is ( - 2, 3).
Check: Substitute -2 for x and 3 for y in
each of the original equations and check for true statements.