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.
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