SOLVING SYSTEMS OF LINEAR EQUATIONS BY ELIMINATION WORKSHEET

Problems 1-7 :Solve each of the following systems of linear equations by elimination.

Problem 1 :

x + y = 2

x - y = 0

Problem 2 :

-3x + 2y = -2

3x - y = -8

Problem 3 :

3x + 5y = 5

x + 5y = -5

Problem 4 :

-2x + y = 14

4x - 3y = -32

Problem 5 :

5x - 6y = 1

7x + 8y = 1

Problem 6 :

2x - 3y = 3

-2x + 3y = 1

Problem 7 :

x - 2y = -5

-7x + 14y = 35

Problem 8 :

Find two numbers such that the sum of them is 18 and the difference between them is 2.

Answers

1. Answer :

x + y = 2 ----A

x - y = 0 ----B

Since we find y in A and -y in B, by adding A and B, y can be eliminated.

A + B :

(x + y) + (x - y) = 2 + 0

x + y + x - y = 2

2x = 2

Divide both sides by 2.

x = 1

Substitute x = 1 into A.

1 + y = 2

Subtract 1 from both sides.

y = 1

The solution is (1, 1).

2. Answer :

-3x + 4y = 6 ----A

3x - y = 3 ----B

Since we find -3x in A and 3x in B, by adding (1) and (2), x can be eliminated.

A + B :

(-3x + 4y) + (3x - y) = 6 + 3

-3x + 4y + 3x - y = 9

3y = 9

Divide botyh sides by 3.

y = 3

Substitute y = 3 into B.

3x - 3 = 3

Add 3 to both sides.

3x = 6

Divide both sides by 3.

x = 2

The solution is (2, 3).

3. Answer :

3x + 5y = 5 ----A

x + 5y = -5 ----B

Multiply both sides of A by -1.

-1(3x + 5y) = -1(5)

-3x - 5y = -5 ----C

Add B and C to eliminate y.

B + C :

(x + 5y) + (-3x - 5y) = (-5) + (-5)

x + 5y - 3x - 5y = -5 - 5

-2x = -10

Divide both sides by -2.

x = 5

Substitute x = 5 into (B).

5 + 5y = -5

Subtract 5 from both sides.

5y = -10

Divide both sides by 5.

y = -2

The solution is (5, -2).

4. Answer :

-2x + y = 14 ----A

4x - 3y = -32 ----B

Multiply A by 2.

2(-2x + y) = 2(14)

-4x + 2y = 28 ----C

Add B and C to eliminate x.

B + C :

(4x - 3y) + (-4x + 2y) = -32 + 28

4x - 3y - 4x + 2y = -4

-y = -4

Multiply both sides by -1.

y = 4.

Substitute y = 4 into A.

-2x + 4 = 14

Subtract 4 from both sides.

-2x = 10

Divide both sides by -2.

x = -5

The solution is (-5, 4).

5. Answer :

5x - 6y = -11 ----A

7x + 8y = 1 ----B

In the above system of linear equations, the number in front of either the variable x or y is not same. We have -6 infront of y in A and 8 in B.

Least common multiple of (6, 8) = 24.

Multiply A by 4 to get -24 infront of y and multiply B by 3 to get 24 infornt of y.

Multiply A by 4.

4(5x - 6y) = 4(-11)

20x - 24y = -44 ----C

Multiply B by 3.

3(7x + 8y) = 3(1)

21x + 24y = 3 ----D

Add C and D to eliminate.

C + D :

(20x - 24y) + (21x + 24y) = -44 + 3

20x - 24y + 21x + 24y = -41

41x = -41

Divide both sides by 41.

x = -1

Substitute x = -1 into B.

7(-1) + 8y = 1

-7 + 8y = 1

Add 7 to both sides.

8y = 8

Divide both sides by 8.

y = 1

The solution is (-1, 1).

6. Answer :

2x - 3y = 3 ----A

-2x + 3y = 1 ----B

Add A and B.

A + B :

(2x - 3y) + (-2x + 3y) = 3 + 1

2x - 3y - 2x + 3y = 4

0 = 4 (false)

In the above step of solving the given system, there is no variable and '0 = 4' is false.

So, the given system of linear equations has NO solution.

7. Answer :

x - 2y = -5 ----A

-7x + 14y = 35 ----B

In B, the coefficients of x, y and the costant term are all evenly divisible by 7. 

So, divide both sides of B by 7.

⁽⁻⁷ˣ ⁺ ¹⁴ʸ⁾⁄₇ = ³⁵⁄₇

⁽⁻⁷ˣ⁾⁄₇ ⁺ ⁽¹⁴ʸ⁾⁄₇ = 5

-x + 2y = 5 ----C

Add A and C.

A + C :

(x - 2y) + (-x + 2y) = -5 + 5

x - 2y - x + 2y = 0

0 = 0 (true)

In the above step of solving the given system, there is no variable and '0 = 0' is true.

So, the given system of linear equations has infinitely many solutions.

8. Answer :

Let x and y be the two numbers

Given : Sum of the two numbers is 18 and the difference between them is 2.

x + y = 18 ----A

x - y = 2 ----B

Add A and B to eliminate y.

(x + y) + (x - y) = 18 + 2

x + y + x - y = 20

2x = 20

Divide both sides by 2.

x = 10

Substitute x = 10 into A.

10 + y = 18

Subtract 10 from both sides.

y = 8

The numbers are 8 and 10.

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