Systems of Equations - Riverside Math - Free Printable

You’ve uploaded a worksheet titled “Solving Systems of Equations – Substitution”, which contains 12 systems of linear equations. Each system is designed to be solved using the substitution method.

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## ✔ What is the Substitution Method?

The substitution method involves:
1. Solving one equation for one variable (e.g., solve for x or y).
2. Substituting that expression into the other equation.
3. Solving the resulting single-variable equation.
4. Plugging the solution back into one of the original equations to find the other variable.
5. Writing the solution as an ordered pair: (x, y).

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Let’s solve all 12 problems step-by-step.

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🔢 Problem 1:

> x = 2y + 1
> 4x - 5y = 13

✔ Already solved for x → substitute into second equation:

→ 4(2y + 1) - 5y = 13
→ 8y + 4 - 5y = 13
→ 3y + 4 = 13
→ 3y = 9
→ y = 3

Now plug into x = 2y + 1:
→ x = 2(3) + 1 = 7

✔ Solution: (7, 3)

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🔢 Problem 2:

> y = -3x - 6
> -5x - 8y = -28

Substitute y into second equation:

→ -5x - 8(-3x - 6) = -28
→ -5x + 24x + 48 = -28
→ 19x + 48 = -28
→ 19x = -76
→ x = -4

Now plug into y = -3x - 6:
→ y = -3(-4) - 6 = 12 - 6 = 6

✔ Solution: (-4, 6)

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🔢 Problem 3:

> y = 2x - 20
> 8x + 7y = 80

Substitute y:

→ 8x + 7(2x - 20) = 80
→ 8x + 14x - 140 = 80
→ 22x = 220
→ x = 10

Then y = 2(10) - 20 = 0

✔ Solution: (10, 0)

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🔢 Problem 4:

> x = -13y - 2
> x = 7y + 18

Both equal x → set them equal:

→ -13y - 2 = 7y + 18
→ -20y = 20
→ y = -1

Plug into either: x = 7(-1) + 18 = -7 + 18 = 11

✔ Solution: (11, -1)

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🔢 Problem 5:

> 2y = 4x → simplify: y = 2x
> -3x + y = -4

Substitute y = 2x:

→ -3x + 2x = -4
→ -x = -4
→ x = 4

Then y = 2(4) = 8

✔ Solution: (4, 8)

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🔢 Problem 6:

> 3x = -y → so y = -3x
> 10x + 4y = -6

Substitute y = -3x:

→ 10x + 4(-3x) = -6
→ 10x - 12x = -6
→ -2x = -6
→ x = 3

Then y = -3(3) = -9

✔ Solution: (3, -9)

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🔢 Problem 7:

> y = x + 16
> 10x + 8y = -88

Substitute y:

→ 10x + 8(x + 16) = -88
→ 10x + 8x + 128 = -88
→ 18x = -216
→ x = -12

Then y = -12 + 16 = 4

✔ Solution: (-12, 4)

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🔢 Problem 8:

> x = 16 - 3y
> 4x + 12y = 96

Substitute x:

→ 4(16 - 3y) + 12y = 96
→ 64 - 12y + 12y = 96
→ 64 = 96 ✘ Contradiction!

This means no solution — the lines are parallel.

✔ Solution: No solution

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🔢 Problem 9:

> x = (1/2)y - 7
> 3y = 42 → so y = 14

Plug y = 14 into first equation:

→ x = (1/2)(14) - 7 = 7 - 7 = 0

✔ Solution: (0, 14)

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🔢 Problem 10:

> x = -y
> -3x - 3y = 0

Substitute x = -y:

→ -3(-y) - 3y = 0
→ 3y - 3y = 0
→ 0 = 0 ✔ Always true!

This means infinitely many solutions — the equations represent the same line.

✔ Solution: Infinitely many solutions

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🔢 Problem 11:

> 4x = 8y - 52 → divide by 4: x = 2y - 13
> 5x - y = 34

Substitute x = 2y - 13:

→ 5(2y - 13) - y = 34
→ 10y - 65 - y = 34
→ 9y = 99
→ y = 11

Then x = 2(11) - 13 = 22 - 13 = 9

✔ Solution: (9, 11)

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🔢 Problem 12:

> 3y = 12 - 6x → divide by 3: y = 4 - 2x
> -15x - 4y = -2

Substitute y = 4 - 2x:

→ -15x - 4(4 - 2x) = -2
→ -15x -16 + 8x = -2
→ -7x = 14
→ x = -2

Then y = 4 - 2(-2) = 4 + 4 = 8

✔ Solution: (-2, 8)

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## 📋 Final Answers Summary:

| Problem | Solution |
|---------|----------------------|
| 1 | (7, 3) |
| 2 | (-4, 6) |
| 3 | (10, 0) |
| 4 | (11, -1) |
| 5 | (4, 8) |
| 6 | (3, -9) |
| 7 | (-12, 4) |
| 8 | No solution |
| 9 | (0, 14) |
| 10 | Infinitely many sol. |
| 11 | (9, 11) |
| 12 | (-2, 8) |

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## 💡 Tips for Students:
- Always look for an equation already solved for a variable — that’s your starting point!
- When you get “0=0”, it means infinite solutions.
- When you get a false statement like “64=96”, it means no solution.
- Double-check your arithmetic — especially with negatives and fractions.

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✔ You’re all set! Let me know if you’d like a printable answer key or step-by-step video explanations for any problem.