The following steps would be useful to solve a system of linear equations in two variables.
Step 1 :
The variable has to be eliminated must have the same number infront of it in both the equations with different signs. If it is not so, multiply one or both the equations by an appropriate number to get the system as said above.
Step 2 :
Add two equations and eliminate one of the variables and solve for the other variable.
Step 3 :
Substitute the value found in step 2 into one of the given two equations and solve for the remaining variable.
Step 4 :
Write the solution in as an ordered pair (x, y).
Solve each of the following systems of linear equations by elimination.
Example 1 :
x - 2y = 12
2x + 2y = 18
Solution :
x - 2y = 12 ----(1)
2x + 2y = 18 ----(2)
Since we find -2y in (1) and 2y in (2), by adding (1) and (2), y can be eliminated.
(1) + (2) :
(x - 2y) + (2x + 2y) = 12 + 18
x - 2y + 2x + 2y = 30
3x = 30
Divide both sides by 3.
³ˣ⁄₃ = ³⁰⁄₃
x = 10
Substitute x = 10 into (1).
10 - 2y = 12
Subtract 10 from both sides.
(10 - 2y) - 10 = 12 - 10
10 -2y - 10 = 2
-2y = 2
Divide both sides by -2.
⁻²ʸ⁄₋₂ = ²⁄₋₂
y = -1
The solution is (10, -1).
Example 2 :
x + 3y = -5
-x + 5y = -11
Solution :
x + 3y = -5 ----(1)
-x + 5y = -11 ----(2)
Since we find x in (1) and -x in (2), by adding (1) and (2), y can be eliminated.
(1) + (2) :
(x + 3y) + (-x + 5y) = (-5) + (-11)
x + 3y - x + 5y = -5 - 11
8y = -16
Divide both sides by 8.
⁸ʸ⁄₈ = ⁻¹⁶⁄₈
y = -2
Substitute y = -2 into (1).
x + 3(-2) = -5
x - 6 = -5
Add 6 to both sides.
(x - 6) + 6 = -5 + 6
x - 6 + 6 = 1
x = 1
The solution is (1, -2).
Example 3 :
6x - 5y = -1
6x + 7y = -13
Solution :
6x - 5y = -1 ----(1)
6x + 7y = -13 ----(2)
Multiply both sides of (1) by -1.
-1(6x - 5y) = -1(-1)
-6x + 5y = 1 ----(3)
Add (2) and (3) to eliminate x.
(6x + 7y) + (-6x + 5y) = -13 + 1
6x + 7y - 6x + 5y = -12
12y = -12
Divide both sides by -12.
¹²ʸ⁄₁₂ = ⁻¹²⁄₁₂
y = -1
Substitute y = -1 in to (1).
6x - 5(-1) = -1
6x + 5 = -1
Subtract 5 from both sides.
(6x + 5) - 5 = -1 - 5
6x + 5 - 5 = -6
6x = -6
Divide both sides by 6.
⁶ˣ⁄₆ = ⁻⁶⁄₆
x = -1
The solution is (-1, -1).
Example 4 :
2x - 9y = -29
5x - 9y = -32
Solution :
2x - 9y = -29 ----(1)
5x - 9y = -32 ----(2)
Multiply both sides of (1) by -1.
-1(2x - 9y) = -1(-29)
-2x + 9y = 29 ----(3)
Add (2) and (3) to eliminate y.
(5x - 9y) + (-2x + 9y) = -32 + 29
5x - 9y - 2x + 9y = -3
3x = -3
Divide both sides by 3.
³ˣ⁄₃ = ⁻³⁄₃
x = -1
Substitute x = -1 into (1).
2(-1) - 9y = -29
-2 - 9y = -29
Add 2 to both sides.
(-2 - 9y) + 2 = -29 + 2
-2 - 9y + 2 = -27
-9y = -27
Divide both sides by -9.
⁻⁹ʸ⁄₃ = ²⁷⁄₃
y = 3
The solution is (-1, 3).
Example 5 :
2x + 9y = 8
4x - 5y = -30
Solution :
2x + 9y = 8 ----(1)
4x - 5y = -30 ----(2)
In the above system of linear equations, multiplying the first equation by -2, we will get the same number in front of x in both the equations with different signs.
Multiply (1) by -2.
-2(2x + 9y) = -2(8)
-4x - 18y = -16 ----(3)
Add (2) and (3) to eliminate x.
(4x - 5y) + (-4x - 18y) = -30 + (-16)
4x - 5y + -4x - 18y = -30 - 16
-23y = -46
Divide both sides by -23.
⁻²³ʸ⁄₋₂₃ = ⁻⁴⁶⁄₋₂₃
y = 2
Substitute y = 2 into (1).
2x + 9(2) = 8
2x + 18 = 8
Subtract 148 from both sides.
2x + 18 - 18 = 8 - 18
2x = -10
Divide both sides by 2.
²ˣ⁄₂ = ⁻¹⁰⁄₂
x = -5
The solution is (-5, 2).
Example 6 :
4x + 5y = 7
6x - 7y = -33
Solution :
4x + 5y = 7 ----(1)
6x - 7y = -33 ----(2)
In the above system of linear equations, the number in front of either the variable x or y is not same. We have 4 infront of x in (1) and 6 in (2).
Least common multiple of (4, 6) = 12.
Multiply (1) by -3 to get -12 infront of x and multiply (2) by 2 to get 12 infornt of x.
Multiply (1) by -3.
-3(4x + 5y) = -3(7)
-12x - 15y = -21 ----(3)
Multiply (2) by 2.
2(6x - 7y) = 2(-33)
12x - 14y = -66 ----(4)
Add (3) and (4) to eliminate x.
(-12x - 15y) + (12x - 14y) = (-21) + (-66)
-12x - 15y + 12x - 14y = -21 - 66
-29y = -87
Divide both sides by -29.
⁻²⁹ʸ⁄₋₂₉ = ⁻⁸⁷⁄₋₂₉
y = 3
Substitute y = 3 into (1).
4x + 5(3) = 7
4x + 15 = 7
Subtract 15 from both sides.
4x + 15 - 15 = 7 - 15
4x = -8
Divide both sides by 4.
⁴ˣ⁄₄ = ⁻⁸⁄₄
x = -2
The solution is (-2, 3).
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