Worksheet for solving systems of linear equations using substitution.

We are given a worksheet titled “Solving Systems of Linear Equations by Substitution”. We’ll solve each system step-by-step using the substitution method, which involves solving one equation for one variable and substituting that expression into the other equation.

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Problem 1:

> -2x + y = 5
> 6x - 3y = -9

Step 1: Solve the first equation for y:
> -2x + y = 5 → y = 2x + 5

Step 2: Substitute into the second equation:
> 6x - 3(2x + 5) = -9
> 6x - 6x - 15 = -9
> 0x - 15 = -9
> -15 = -9 → Contradiction

✔ Answer: No solution (The lines are parallel)

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Problem 2:

> y = x + 4
> 3x - 4y = -19

Step 1: Already solved for y: y = x + 4

Step 2: Substitute into second equation:
> 3x - 4(x + 4) = -19
> 3x - 4x - 16 = -19
> -x - 16 = -19
> -x = -3 → x = 3

Step 3: Plug back to find y:
> y = 3 + 4 = 7

✔ Answer: (3, 7)

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Problem 3:

> -4x + y = 6
> -5x - y = 21

Step 1: Solve first equation for y:
> y = 4x + 6

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

Step 3: Plug back to find y:
> y = 4(-3) + 6 = -12 + 6 = -6

✔ Answer: (-3, -6)

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Problem 4:

> 6x + 4y = -8
> -6x + y = 28

Step 1: Solve second equation for y:
> y = 6x + 28

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

Step 3: Plug back to find y:
> y = 6(-4) + 28 = -24 + 28 = 4

✔ Answer: (-4, 4)

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Problem 5:

> x = -2y
> x - 9 = 9

Step 1: Solve second equation:
> x = 9 + 9 = 18

Step 2: Plug into first equation:
> 18 = -2y → y = -9

✔ Answer: (18, -9)

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Problem 6:

> x + y = 3
> x - y = -3

Step 1: Solve first equation for x:
> x = 3 - y

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

Step 3: Plug back to find x:
> x = 3 - 3 = 0

✔ Answer: (0, 3)

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Problem 7:

> x = y - 4
> -2x + 3y = 6

Step 1: Already solved for x: x = y - 4

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

Step 3: Plug back to find x:
> x = -2 - 4 = -6

✔ Answer: (-6, -2)

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Problem 8:

> -5x + 20 = 5y
> -2x + y = 7

Step 1: Solve second equation for y:
> y = 2x + 7

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

Step 3: Plug back to find y:
> y = 2(-1) + 7 = -2 + 7 = 5

✔ Answer: (-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 these graphed or solved by elimination too!