Worksheet for solving systems of linear equations using the substitution method.

Let’s solve each system of linear equations using the substitution method, as instructed.

---

Problem 1:

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

✔ Step 1: Solve Equation (1) for `y`:
```
-2x + y = 5
→ y = 2x + 5
```

✔ Step 2: Substitute `y = 2x + 5` into Equation (2):
```
6x - 3(2x + 5) = -9
6x - 6x - 15 = -9
0x - 15 = -9
-15 = -9 → ✘ Contradiction!
```

➡️ Conclusion: No solution. The lines are parallel.

---

Problem 2:

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

✔ Step 1: Already solved for `y` in (1): `y = x + 4`

✔ Step 2: Substitute into (2):
```
3x - 4(x + 4) = -19
3x - 4x - 16 = -19
-x - 16 = -19
-x = -3
→ x = 3
```

✔ Step 3: Plug back into `y = x + 4`:
```
y = 3 + 4 = 7
```

✔ Answer: `(3, 7)`

---

Problem 3:

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

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

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

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

✔ Answer: `(-3, -6)`

---

Problem 4:

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

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

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

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

✔ Answer: `(-4, 4)`

---

Problem 5:

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

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

✔ Step 2: Plug into (1):
```
18 = -2y
→ y = -9
```

✔ Answer: `(18, -9)`

---

Problem 6:

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

✔ 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 into `x = 3 - y`:
```
x = 3 - 3 = 0
```

✔ Answer: `(0, 3)`

---

Problem 7:

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

✔ Step 1: Already solved for `x` in (1): `x = y - 4`

✔ Step 2: Substitute into (2):
```
-2(y - 4) + 3y = 6
-2y + 8 + 3y = 6
y + 8 = 6
→ y = -2
```

✔ Step 3: Plug into `x = y - 4`:
```
x = -2 - 4 = -6
```

✔ Answer: `(-6, -2)`

---

Problem 8:

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

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

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

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

✔ Answer: `(-1, 5)`

---

## ✔ 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!