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 .
Example
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
y.
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
equation.
2x + y =  1 
Use the second equation. 
2(1  y) + y = 1 
Substitute 1  y for x. 
2  2y + y = 1 
Distribute. 
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.
