Elimination Method:
- Obtain coefficients of x(or y) that differ only in sign by multiplying all terms of one or both equations by suitably chosen constants.
- Add the equations to eliminate one variable, and solve the resulting equation.
- Back substituting the value obtained in step 2 into either of the original equations and solve for the other variable.
- Check the solution in both of the original equations.
Intersecting Lines:
Example:
Solve the system of linear equations:
3x + 2y = 4 ----- Equation 1
5x - 2y = 8 ------- Equation 2
Solution:
Adding both equations, to eliminate y, we get
3x + 2y = 4 -----equation 1
5x - 2y = 8 ------- equation 2.
----------------------Adding equations.
8x = 12
x = 12/8 = 3/2. By back substituting we can solve for y.
substituting the value of x in equation 1 we have
3x + 2y = 4
3(3/2) + 2y = 4
y = -1/4
The solution of the two equations is x = 3/2 and y = -1/4
Example:
Solve the system of linear equations:
2x - 3y = -7 ----- Equation 1
3x + y = -5 ------- Equation 2
Solution:
Here we can obtain the coefficients that differ only in sign by multiplying equation 2 by 3.
2x - 3y = -7 -----equation 1
3( 3x + y = -5) ------- equation 2.
The equations we get are:
2x - 3y = -7 -----equation 1
9x + 3y = -15 ------- equation 2. Adding both equations, to eliminate y, we get
----------------------
11x = -22
x = -2. By back substituting we can solve for y.
substituting the value of x in equation 1 we have
2x - 3y = -7 -----equation 1
2(-2) - 3y = -7
-4 - 3y = -7
- 3y = -7 + 4
- 3y = -3
y = 1
The solution of the two equations is x = -2 and y = 1
Verification:
Substituting the value of x and y in one of the two equations:
2x - 3y = -7 -----equation 1
2x - 3y
2(-2) - 3(1)
-4 - 3
- 7
Substituting in the equation 2:
3x + y = -5 ------- equation 2.
3(-2) + 1
-6 + 1
-5
Hence the values of x and y are correct.
Example:
Solve the system of linear equations:
5x + 3y = 9 ----- Equation 1
2x - 4y = 14 ------- Equation 2
Solution:
Here we can obtain the coefficients that differ only in sign by multiplying equation 1 by 4 and equation 2 by 3.
4(5x + 3y = 9) -----equation 1
3(2x - 4y = 14) ------- equation 2.
The equations we get are:
20x + 12y = 36 -----equation 1
6x - 12y = 42 ------- equation 2. Adding both equations, to eliminate y, we get
----------------------
26x = 78
x = 3. By back substituting we can solve for y.
substituting the value of x in equation 1 we have
5x + 3y = 9 -----equation 1
5(3) + 3y = 9
15 + 3y = 9
3y = 9 - 15
3y = -6
y = -2
The solution of the two equations is x = 3 and y = -2
Verification:
Substituting the value of x and y in one of the two equations:
5x + 3y = 9 -----equation 1
5(3) + 3(-2)
15 + -6
9
Hence the values of x and y are correct.
Parallel Lines:
Method of Elimination: No-solution Case
Example:
Solve the system of linear equations:
x - 2y = 3 ----- Equation 1
-2x + 4y = 1 ------- Equation 2
Solution:
Here we obtain the coefficients that differ only in sign by multiplying equation 1 by 2 .
2(x - 2y = 3) ----- Equation 1
-2x + 4y = 1 ------- Equation 2
The equations we get are:
2x - 4y = 6 ----- Equation 1
-2x + 4y = 1 ------- Equation 2
Adding both equations, to eliminate y, we get
----------------------
0 = 7 ------ False statement
Since there are no values of x and y for which 0 = 7 we can conclude that the system is inconsistent and has no solution.
In other words the two lines do not intersect they are parallel lines
Two lines Coincide:
Method of Elimination: Many-solution Case
Example:
Solve the system of linear equations:
2x - y = 2 ----- Equation 1
4x - 2y = 2 ------- Equation 2
Solution:
To obtain the coefficients that differ only in sign by multiplying equation 2 by -1/2 .
The equations we get are:
2x - y = 2 ----- Equation 1
-1/2( 4x -2y = 2) ------- Equation 2
-2x + y = -1 Adding both equations, to eliminate y, we get
----------------------
0 = 0
Since the two equations turn out to be equivalent or have the same solution set, the system has infinitely many solutions.
In other words the two lines coincide
Directions: Solve for the variables using the method of Elimination. Also write at least 5 examples of your own.
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