The following steps will be useful to solve the systems of linear equations using substitution.
Step 1 :
In the given two equations, solve one of the equations either for x or y.
Step 2 :
Substitute the result of step 1 into other equation and solve for the second variable.
Step 3 :
Using the result of step 2 and step 1, solve for the first variable.
Solve the following system of linear equations using substitution method.
Example 1 :
x = 8 – 2y
2x + 3y = 13
Solution :
x = 8 – 2y ----(1)
2x + 3y = 13 ----(2)
(1) is solved for x.
Substitute x = 8 – 2y in (2),
2(8 – 2y) + 3y = 13
16 – 4y + 3y = 13
16 – y = 13
y = 3
Substitute y = 3 in (1).
x = 8 – 2(3)
x = 2
Hence, the solution is (x, y) = (2, 3).
Example 2 :
y = 4 + x
5x - 3y = 0
Solution :
y = 4 + x ----(1)
5x - 3y = 0 ----(2)
(1) is solved for y.
Substitute y = 4 + x in (2).
5x – 3(4 + x) = 0
5x – 12 – 3x = 0
2x – 12 = 0
x = 6
Substitute x = 6 in (1).
y = 4 + 6
y = 10
Hence, the solution is (x, y) = (6, 10).
Example 3 :
x = -10 – 2y
3y – 2x = -22
Solution :
x = -10 – 2y ----(1)
3y – 2x = -22 ----(2)
(1) is solved for x.
Substitute x = -10 – 2y in (2).
3y – 2(-10 – 2y) = -22
3y + 20 + 4y = -22
7y + 20 = -22
7y = -22 – 20
7y = -42
y = -6
Substitute y = -6 in (1).
x = -10 – 2(- 6)
x = -10 + 12
x = 2
Hence, the solution is (x, y) = (2, -6).
Example 4 :
x = -1 + 2y
x = 9 – 2y
Solution :
x = -1 + 2y ----(1)
x = 9 – 2y ----(2)
(1) and (2) are solved for x.
(1) = (2)
x = x
-1 + 2y = 9 – 2y
-1 + 4y = 9
4y = 10
y = 5/2
y = 2.5
Substitute y = 2.5 in (2).
x = 9 – 2(2.5)
x = 9 – 5
x = 4
Hence, the solution is (x, y) = (4, 2.5).
Example 5 :
3x – 2y = 8
x = 3y + 12
Solution :
3x – 2y = 8 ----(1)
x = 3y + 12 ----(2)
(2) is solved for x.
Substitute x = 3y + 12 in (1).
3(3y + 12) – 2y = 8
9y + 36 – 2y = 8
7y + 36 = 8
7y = -28
y = -4
Substitute y = -4 in (2).
x = 3(- 4) + 12
x = -12 + 12
x = 0
Hence, the solution is (x, y) = (0, -4).
Example 6 :
x + 2y = 8
y = 7 – 2x
Solution :
x + 2y = 8 ----(1)
y = 7 – 2x ----(2)
(2) is solved for y.
Substitute y = 7 – 2x in (1).
x + 2(7 – 2x) = 8
x + 14 – 4x = 8
-3x = 8 – 14
x = 2
Substitute x = 2 in (2).
y = 7 – 2(2)
y = 3
Hence, the solution is (x, y) = (2, 3).
Example 7 :
x = -1 – 2y
2x – 3y = 12
Solution :
x = -1 – 2y ----(1)
2x – 3y = 12 ----(2)
(1) is solved for x.
Substitute x = -1 – 2y in (2).
2(-1 – 2y) – 3y = 12
-2 - 4y – 3y = 12
-7y = 12 + 2
y = -2
Substitute y = -2 in (1).
x = -1 – 2y
x = -1 – 2(-2)
x = 3
Hence, the solution is (x, y) = (3, -2).
Example 8 :
y = 5x
7x – 2y = 3
Solution :
y = 5x ----(1)
7x – 2y = 3 ----(2)
(1) is solved for y.
Substitute y = 5x in (2).
7x – 2(5x) = 3
7x – 10x = 3
x = -1
Substitute x = -1 in equation (1).
y = -5
Hence, the solution is (x, y) = (-1, -5).
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