Systems of Linear Equations Worksheets with Answer Key - Free Printable

Sure! Let’s solve each system of linear equations by substitution, as requested. The substitution method involves solving one equation for one variable and substituting that expression into the other equation.

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Problem 1:

Equations:
- -2x + y = 5 → (1)
- 6x - 3y = -9 → (2)

Step 1: Solve equation (1) for y:
> y = 2x + 5

Step 2: Substitute into equation (2):
> 6x - 3(2x + 5) = -9
> 6x - 6x - 15 = -9
> -15 = -9 → Contradiction

✔ No solution — the lines are parallel.

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Problem 2:

Equations:
- y = x + 4 → (1)
- 3x - 4y = -19 → (2)

Step 1: Already solved for y in (1). Substitute into (2):
> 3x - 4(x + 4) = -19
> 3x - 4x - 16 = -19
> -x - 16 = -19
> -x = -3
> x = 3

Step 2: Plug x = 3 into (1):
> y = 3 + 4 = 7

✔ Solution: (3, 7)

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Problem 3:

Equations:
- -4x + y = 6 → (1)
- -5x - y = 21 → (2)

Step 1: Solve (1) for y:
> y = 4x + 6

Step 2: Substitute into (2):
> -5x - (4x + 6) = 21
> -5x - 4x - 6 = 21
> -9x = 27
> x = -3

Step 3: Plug x = -3 into y = 4x + 6:
> y = 4(-3) + 6 = -12 + 6 = -6

✔ Solution: (-3, -6)

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Problem 4:

Equations:
- 6x + 4y = -8 → (1)
- -6x + y = 28 → (2)

Step 1: Solve (2) for y:
> y = 6x + 28

Step 2: Substitute into (1):
> 6x + 4(6x + 28) = -8
> 6x + 24x + 112 = -8
> 30x = -120
> x = -4

Step 3: Plug x = -4 into y = 6x + 28:
> y = 6(-4) + 28 = -24 + 28 = 4

✔ Solution: (-4, 4)

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Problem 5:

Equations:
- x = -2y → (1)
- x - 9 = 9 → (2)

Step 1: Solve (2) for x:
> x = 9 + 9 = 18

Step 2: Plug x = 18 into (1):
> 18 = -2y
> y = -9

✔ Solution: (18, -9)

*(Note: You could also substitute (1) into (2): -2y - 9 = 9 → -2y = 18 → y = -9, then x = 18)*

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Problem 6:

Equations:
- x + y = 3 → (1)
- x - y = -3 → (2)

Step 1: Solve (1) for x:
> x = 3 - y

Step 2: Substitute into (2):
> (3 - y) - y = -3
> 3 - 2y = -3
> -2y = -6
> y = 3

Step 3: Plug y = 3 into x = 3 - y:
> x = 3 - 3 = 0

✔ Solution: (0, 3)

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Problem 7:

Equations:
- x = y - 4 → (1)
- -2x + 3y = 6 → (2)

Step 1: Substitute (1) into (2):
> -2(y - 4) + 3y = 6
> -2y + 8 + 3y = 6
> y + 8 = 6
> y = -2

Step 2: Plug y = -2 into (1):
> x = -2 - 4 = -6

✔ Solution: (-6, -2)

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Problem 8:

Equations:
- -5x + 20 = 5y → (1)
- -2x + y = 7 → (2)

Step 1: Solve (2) for y:
> y = 2x + 7

Step 2: Substitute into (1):
> -5x + 20 = 5(2x + 7)
> -5x + 20 = 10x + 35
> -5x - 10x = 35 - 20
> -15x = 15
> x = -1

Step 3: Plug x = -1 into y = 2x + 7:
> y = 2(-1) + 7 = -2 + 7 = 5

✔ Solution: (-1, 5)

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## ✔ Final Answers:

1. No solution
2. (3, 7)
3. (-3, -6)
4. (-4, 4)
5. (18, -9)
6. (0, 3)
7. (-6, -2)
8. (-1, 5)

Let me know if you’d like to see graphing or elimination methods too!