Elimination of Systems of Linear Equation

1. Solving Systems
of Linear Equation
by Elimination

2. Solving Systems of Linear Equation by Elimination
Solving system of linear equations by elimination uses either addition or
subtraction in order to eliminate any of the variables one at a time. If one
of the variables is eliminated, the whole equation will be transformed
into a linear equation in one variable. Eventually, the value of one of the
variables will be obtained and will be used in finding the value of the
other variable.

3. Solving Systems of Linear Equation by Elimination
Solve the system:
5x + y = 0 (1st equation)
-3x -2y = 7 (2nd equation)
Solution:
Eliminate y by multiplying both sides of the first equation by 2 to obtain
another equation.
2(5x + y) = 0(2)
10x + 2y = 0 (3rd equation)
Ex. 1

4. Solving Systems of Linear Equation by Elimination
Solve the system:
5x + y = 0 (1st equation)
-3x -2y = 7 (2nd equation)
10x + 2y = 0 (3rd equation)
Solution:
Add the the third and second equation to get the value of x.
10x + 2y = 0
-3x + 2y = 7
7x = 7
7x/7 = 7/7
x = 1
Ex. 1

5. Solving Systems of Linear Equation by Elimination
Solve the system:
5x + y = 0 (1st equation)
-3x -2y = 7 (2nd equation)
10x + 2y = 0 (3rd equation)
Solution:
Substitute 1 for x in the first equation to get the value of y.
5x + y = 0
5(1) + y = 0
y = 0 - 5
y = -5
Ex. 1

6. Solving Systems of Linear Equation by Elimination
Solve the system:
5x + y = 0 (1st equation)
-3x -2y = 7 (2nd equation)
Thus:
The solution set of the system is (1, -5).
Ex. 1

7. Solving Systems of Linear Equation by Elimination
Checking:
5x + y = 0 (1st equation)
-3x -2y = 7 (2nd equation)
Substitute (1,-5) in the given equations.
Ex. 1
5x + y = 0
5(1) + (-5) = 0
5 - 5 = 0
0 = 0
-3x - 2y = 7
-3(1) -2(-5) = 7
-3 + 10 = 7
7 = 7

9. Solving Systems of Linear Equation by Elimination
Solve the system:
4x - 3y = -1 (1st equation)
5x - 2y = 4 (2nd equation)
Solution:
To eliminate the variable y, multiply both sides of the first equation by
-2, and both sides of the second equation by 3. Then add the two new
equations.
Ex. 2
(-2)4x - 3y = -1(-2)
-8x + 6y = 2 (3rd eq.)
(3)5x - 2y = 4(3)
15x -6y = 12 (4th eq.)

10. Solving Systems of Linear Equation by Elimination
Solve the system:
4x - 3y = -1 (1st equation)
5x - 2y = 4 (2nd equation)
-8x + 6y = 2 (3rd eq.)
15x -6y = 12 (4th eq.)
Solution:
Add the the third and fourth equation to get the value of x.
-8x + 6y = 2
15x - 6y = 12
7x = 14
7x/7 = 14/7
x = 2
Ex. 2

11. Solving Systems of Linear Equation by Elimination
Solve the system:
4x - 3y = -1 (1st equation)
5x - 2y = 4 (2nd equation)
Solution:
Substitute the value of x in either of the equations to solve for y.
4x - 3y = -1
4(2) - 3y = -1
8 - 3y = -1
-3y = -1 - 8
-3y = -9
-3y/-3 = -9/-3
y= 3
Ex. 2

12. Solving Systems of Linear Equation by Elimination
Solve the system:
4x - 3y = -1 (1st equation)
5x - 2y = 4 (2nd equation)
Thus:
The solution set of the system is (2, 3).
Ex. 2

13. Solving Systems of Linear Equation by Elimination
Checking:
4x - 3y = -1 (1st equation)
5x - 2y = 4 (2nd equation)
Substitute (2, 3) in the given equations.
Ex. 2
4x - 3y = -1
4(2) - 3(3) = -1
8 - 9 = -1
-1 = -1
5x - 2y = 4
5(2) - 2(3) = 4
10 - 6 = 4
4 = 4