Solving Linear Systems of Equations by Elimination
Solving Linear Systems of Equations by Elimination
Example
Use elimination to find the solution of this system.
x - 3y = -17 First equation
-x + 8y = 52 Second equation
Solution
Add the two equations.
x
- x
-
+
3y
8y
=
=
-
17
52
0x
+
5y
=
35
Simplify. The x-terms have been eliminated.
To solve for y, divide both sides by 5.
To find the value of x, substitute 7 for y in either of
the original equations. Then solve for x.
5y
y
= 35
= 7
We will use the first equation.
Substitute 7 for y.
Multiply.
Add 21 to both sides.
The solution of the system is (4, 7).
To check the solution, substitute 4 for x and 7 for y into
each original equation. Then simplify.
In each case, the result will be a true statement.
The details of the check are left to you.
x - 3y
x - 3(7)
x - 21
x
= -17
= -17
= -17
= 4
In the two original equations in the previous
example, the coefficients of x were opposites.
Thus, when the equations were added, the
x-terms were eliminated.
1x
- 1x
-
+
3y
8y
=
=
-
17
52
5y
=
35
When the coefficients of neither variable are opposites, we choose a
variable. Then we multiply both sides of one (or both) equations by an
appropriate number (or numbers) to make the coefficients of that variable
opposites.
Note:
The Multiplication Principle of Equality
enables us to multiply both sides of an
equation by the same nonzero number
without changing the solutions of the
equation.
