Comprehensive practice sheet featuring 12 problems designed to help students master solving systems of linear equations using the elimination method.

Let’s solve each system of equations by elimination. We’ll go one at a time, carefully adding or subtracting the equations to eliminate one variable, then solving for the other.

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Problem 1:
-4x - 2y = -12
4x + 8y = -24

Add the two equations together (the x terms will cancel):

(-4x + 4x) + (-2y + 8y) = -12 + (-24)
0x + 6y = -36
→ 6y = -36
→ y = -6

Now plug y = -6 into one of the original equations. Let’s use the first:

-4x - 2(-6) = -12
-4x + 12 = -12
-4x = -24
x = 6

✔ Solution: (6, -6)

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Problem 2:
4x + 8y = 20
-4x + 2y = -30

Add them (x cancels again):

(4x - 4x) + (8y + 2y) = 20 + (-30)
0x + 10y = -10
→ 10y = -10
→ y = -1

Plug into first equation:

4x + 8(-1) = 20
4x - 8 = 20
4x = 28
x = 7

✔ Solution: (7, -1)

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Problem 3:
x - y = 11
2x + y = 19

Add them (y cancels):

(x + 2x) + (-y + y) = 11 + 19
3x + 0y = 30
→ 3x = 30
→ x = 10

Plug into first equation:

10 - y = 11
-y = 1
y = -1

✔ Solution: (10, -1)

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Problem 4:
-6x + 5y = 1
6x + 4y = -10

Add them (x cancels):

(-6x + 6x) + (5y + 4y) = 1 + (-10)
0x + 9y = -9
→ 9y = -9
→ y = -1

Plug into second equation:

6x + 4(-1) = -10
6x - 4 = -10
6x = -6
x = -1

✔ Solution: (-1, -1)

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Problem 5:
-2x - 9y = -25
-4x - 9y = -23

Subtract the first equation from the second to eliminate y:

(-4x - 9y) - (-2x - 9y) = -23 - (-25)
-4x - 9y + 2x + 9y = -23 + 25
-2x = 2
→ x = -1

Plug into first equation:

-2(-1) - 9y = -25
2 - 9y = -25
-9y = -27
y = 3

✔ Solution: (-1, 3)

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Problem 6:
8x + y = -16
-3x + y = -5

Subtract the second equation from the first to eliminate y:

(8x + y) - (-3x + y) = -16 - (-5)
8x + y + 3x - y = -16 + 5
11x = -11
→ x = -1

Plug into first equation:

8(-1) + y = -16
-8 + y = -16
y = -8

✔ Solution: (-1, -8)

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Problem 7:
-6x + 6y = 6
-6x + 3y = -12

Subtract the second equation from the first:

(-6x + 6y) - (-6x + 3y) = 6 - (-12)
-6x + 6y + 6x - 3y = 6 + 12
3y = 18
→ y = 6

Plug into first equation:

-6x + 6(6) = 6
-6x + 36 = 6
-6x = -30
x = 5

✔ Solution: (5, 6)

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Problem 8:
7x + 2y = 24
8x + 2y = 30

Subtract first from second:

(8x + 2y) - (7x + 2y) = 30 - 24
8x + 2y - 7x - 2y = 6
x = 6

Plug into first equation:

7(6) + 2y = 24
42 + 2y = 24
2y = -18
y = -9

✔ Solution: (6, -9)

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Problem 9:
5x + y = 9
10x - 7y = -18

Multiply first equation by 7 so we can eliminate y:

7*(5x + y) = 7*9 → 35x + 7y = 63

Now add to second equation:

35x + 7y = 63
10x - 7y = -18
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45x + 0y = 45
→ 45x = 45
→ x = 1

Plug into first equation:

5(1) + y = 9
5 + y = 9
y = 4

✔ Solution: (1, 4)

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Problem 10:
-4x + 9y = 9
x - 3y = -6

Multiply second equation by 3 to match coefficients of y:

3*(x - 3y) = 3*(-6) → 3x - 9y = -18

Now add to first equation:

-4x + 9y = 9
3x - 9y = -18
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-x + 0y = -9
→ -x = -9
→ x = 9

Plug into second equation:

9 - 3y = -6
-3y = -15
y = 5

✔ Solution: (9, 5)

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Problem 11:
-3x + 7y = -16
-9x + 5y = 16

Multiply first equation by 3 to match x-coefficients:

3*(-3x + 7y) = 3*(-16) → -9x + 21y = -48

Now subtract second equation from this:

(-9x + 21y) - (-9x + 5y) = -48 - 16
-9x + 21y + 9x - 5y = -64
16y = -64
→ y = -4

Plug into first equation:

-3x + 7(-4) = -16
-3x - 28 = -16
-3x = 12
x = -4

✔ Solution: (-4, -4)

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Problem 12:
-7x + y = -19
-2x + 3y = -19

Multiply first equation by 3 to match y-coefficients:

3*(-7x + y) = 3*(-19) → -21x + 3y = -57

Now subtract second equation from this:

(-21x + 3y) - (-2x + 3y) = -57 - (-19)
-21x + 3y + 2x - 3y = -57 + 19
-19x = -38
→ x = 2

Plug into first equation:

-7(2) + y = -19
-14 + y = -19
y = -5

✔ Solution: (2, -5)

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Final Answer:
1) (6, -6)
2) (7, -1)
3) (10, -1)
4) (-1, -1)
5) (-1, 3)
6) (-1, -8)
7) (5, 6)
8) (6, -9)
9) (1, 4)
10) (9, 5)
11) (-4, -4)
12) (2, -5)