Solving Linear Systems of Equations by Substitution
Graphing is not always the best way to find the solution of a system of
equations. It may be difficult to read the coordinates of the point of
intersection. This is especially true when the coordinates are not integers.
Instead we can use algebraic methods to solve the system. One algebraic
method for finding the solution of a linear system is the substitution
method.
Procedure —
To Solve a Linear System By Substitution
Step 1 Solve one equation for one of the variables in terms of the
other variable.
Step 2 Substitute the expression found in Step 1 into the other
equation. Then, solve for the variable.
Step 3 Substitute the value obtained in Step 2 into one of the
equations containing both variables. Then, solve for the
remaining variable.
Step 4 To check the solution, substitute it into each original
equation. Then simplify.
Example
Use substitution to find the solution of this system.
2x + y = 4 First equation
3x + y = 7 Second equation
Solution
Step 1 Solve one equation for one of the variables in terms of the other
variable.
| Either equation may be solved for either variable.
For instance, let’s solve the first equation for y.
Subtract 2x from both sides.
The equation y = -2x + 4 means that y and
the expression -2x + 4 are equivalent.
|
2x + y
y |
= 4
= -2x + 4 |
Step 2 Substitute the expression found in Step 1 into the other equation.
Then, solve for the variable.
| Substitute -2x + 4 for y in the second equation.
Combine like terms.
Subtract 4 from both sides.
Now we know x = 3.
Next, we will find y. |
3x + y 3x + (-2x + 4)
x + 4
x |
= 7 = 7
= 7
=3 |
Step 3 Substitute the value obtained in Step 2 into one of the equations
containing both variables. Then, solve for the remaining
variable.
| We will use the equation from Step 1.
Substitute 3 for x.
Simplify.
The solution of the system is x = 3 and y = -2.
The solution may also be written as (3, -2). |
y y
y |
= -2x + 4 = -2(3) + 4
= -2 |
Step 4 To check the solution, substitute it into each original equation.
Then simplify.Substitute x = 3 and y = -2 into both of the original equations:
| |
First equation |
|
Second equation |
| Is
Is |
2x
2(3)
6 |
+
+
- |
y
(-2)
2
4 |
= 4 = 4 ?
= 4 ?
= 4 ? Yes |
Is
Is
Is |
3x
3(3)
9 |
+
+
- |
y
(-2)
2
7 |
= 7 = 7 ?
= 7 ?
= 7 ? Yes |
Since (3, -2) satisfies both equations, it is the solution of the system.
Note:
If we graphed the system, the lines would
intersect at the point (3, -2).
|
