HOW TO SOLVE A SYSTEM OF LINEAR EQUATIONS BY ELIMINATION

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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