ELIMINATION - using algebra to solve a linear system
* If the coefficients of the same variable in both equations is equal or of opposite
value (ex. +3 & -3), you can eliminate that variable by adding or subtracting
(multiplying by -1 then adding) the equations. The result is a new equation with
one variable which can be solved.
* If the coefficients of a selected variable are not equal or opposite they must be
made so. This is achieved by multiplying all the terms of an equation by the same
number.
Steps to solve a linear system:
1)
Express both equations in the form ax + by = c
2)
Choose a variable to eliminate. If necessary multiply each equation by a number that
gives the same coefficients - with opposite signs - for that variable in both equations.
3)
Add (or subtract) like terms in the equations to eliminate the chosen variable.
4)
Solve the new equation.
5)
Find the value of the other variable by substituting the number you have found into one of
the original equations and solve. Now you have an ordered pair.
6)
Check that the calculated coordinates work in both original equations.
Example: Solve using elimination.
1)
2)
3x + 2y - 13 = 0
-2x + 4y - 2 = 0
- Step 1
Express equations in ax + by = c form.
1)
2)
3x + 2y = 13
-2x + 4y = 2
- Step 2
Multiply 1) by -2 to make the y coefficients the same but with opposite sign in both equations.
1)X-2 -6x - 4y = -26
- Step 3
Add 2) to 1) to eliminate the y variable.
+
2)
1)
-6x - 4y = -26
-2x + 4y = 2
-8x
= -24
x=3
- Step 4
Substitute x = 3 into equation 1) or 2) and solve for y.
3)
* You can choose any of the
equations.
3x + 2y = 13
3(3) + 2y = 13
9 + 2y = 13
2y = 13 - 9
2y = 4
y=2
The solution is x = 3, y = 2 or (3,2)
Step 5
Check your solution.
1)
3x + 2y - 13 = 0
3(3) + 2(2) - 13 = 0
9 + 4 - 13 = 0
0 = 0 True!!!
2)
-2x + 4y - 2 = 0
-2(3) + 4(2) - 2 = 0
-6 + 8 - 2 = 0
0 = 0 True!!!
Notes:
a) When the coefficients of a variable are exactly the same, you can subtract (or multiply by -1
and add) the two equations to eliminate the variable
b) When the coefficients have the same value but opposite signs, add the two equations to
eliminate the variable
c) If the coefficients are not equal or opposite - multiply the equations by a number which makes
them so.