Question

Determine if these lines are parallel perpendicular or neither

2x + 7y = 28
7X - 2y = 4

Y = -5x + 1
X - 5y = 30

3x + 2y = 8
2x + 3y = -12

Y = -4x - 1
8x + 2y = 14

X + y = 7
X - y = 9

Y = 1/3 x + 9
X - 3y = 3

4x + 9y = 18
Y = 4x + 9

5x - 10y = 20
Y = -2x + 6

-9x + 12y = 24
Y = 3/4 x - 5

Y = x - 3
X - y = 8

10x + 8y = 16
5y = 4x - 15

Y = 5/3 x + 7
6x - 10 y = 10

X - 2y = 18
2x + y = 6

X = 4
X = -6

X = 1
Y = -8

Answer 1

To determine if the given lines are parallel, perpendicular, or neither, we can compare their slopes. Two lines are:

- **Parallel** if they have the same slope.
- **Perpendicular** if the product of their slopes is -1.
- **Neither** if they do not fit into either category.

Let's analyze each pair of equations:

1. **2x + 7y = 28** and **7x - 2y = 4**
- 2x + 7y = 28 → 7y = -2x + 28 → y = (-2/7)x + 4 (slope = -2/7)
- 7x - 2y = 4 → -2y = -7x + 4 → y = (7/2)x - 2 (slope = 7/2)
- Since (-2/7) * (7/2) = -1, these lines are **perpendicular**.

2. **y = -5x + 1** and **x - 5y = 30**
- y = -5x + 1 (slope = -5)
- x - 5y = 30 → -5y = -x + 30 → y = (1/5)x - 6 (slope = 1/5)
- Since (-5) * (1/5) ≠ -1, these lines are **neither**.

3. **3x + 2y = 8** and **2x + 3y = -12**
- 3x + 2y = 8 → 2y = -3x + 8 → y = (-3/2)x + 4 (slope = -3/2)
- 2x + 3y = -12 → 3y = -2x - 12 → y = (-2/3)x - 4 (slope = -2/3)
- Since (-3/2) * (-2/3) ≠ -1, these lines are **neither**.

4. **y = -4x - 1** and **8x + 2y = 14**
- y = -4x - 1 (slope = -4)
- 8x + 2y = 14 → 2y = -8x + 14 → y = -4x + 7 (slope = -4)
- Since both slopes are -4, these lines are **parallel**.

5. **x + y = 7** and **x - y = 9**
- x + y = 7 → y = -x + 7 (slope = -1)
- x - y = 9 → -y = -x + 9 → y = x - 9 (slope = 1)
- Since (-1) * (1) ≠ -1, these lines are **neither**.

6. **y = 1/3 x + 9** and **x - 3y = 3**
- y = 1/3 x + 9 (slope = 1/3)
- x - 3y = 3 → -3y = -x + 3 → y = (1/3)x - 1 (slope = 1/3)
- Since both slopes are 1/3, these lines are **parallel**.

7. **4x + 9y = 18** and **y = 4x + 9**
- 4x + 9y = 18 → 9y = -4x + 18 → y = (-4/9)x + 2 (slope = -4/9)
- y = 4x + 9 (slope = 4)
- Since (-4/9) * (4) ≠ -1, these lines are **neither**.

8. **5x - 10y = 20** and **y = -2x + 6**
- 5x - 10y = 20 → -10y = -5x + 20 → y = (1/2)x - 2 (slope = 1/2)
- y = -2x + 6 (slope = -2)
- Since (1/2) * (-2) = -1, these lines are **perpendicular**.

9. **-9x + 12y = 24** and **y = 3/4 x - 5**
- -9x + 12y = 24 → 12y = 9x + 24 → y = (3/4)x + 2 (slope = 3/4)
- y = (3/4)x - 5 (slope = 3/4)
- Since both slopes are 3/4, these lines are **parallel**.

10. **y = x - 3** and **x - y = 8**
- y = x - 3 (slope = 1)
- x - y = 8 → -y = -x + 8 → y = x - 8 (slope = 1)
- Since both slopes are 1, these lines are **parallel**.

11. **10x + 8y = 16** and **5y = 4x - 15**
- 10x + 8y = 16 → 8y = -10x + 16 → y = (-5/4)x + 2 (slope = -5/4)
- 5y = 4x - 15 → y = (4/5)x - 3 (slope = 4/5)
- Since (-5/4) * (4/5) ≠ -1, these lines are **neither**.

12. **y = 5/3 x + 7** and **6x - 10y = 10**
- y = (5/3)x + 7 (slope = 5/3)
- 6x - 10y = 10 → -10y = -6x + 10 → y = (3/5)x - 1 (slope = 3/5)
- Since (5/3) * (3/5) ≠ -1, these lines are **neither**.

13. **x - 2y = 18** and **2x + y = 6**
- x - 2y = 18 → -2y = -x + 18 → y = (1/2)x - 9 (slope = 1/2)
- 2x + y = 6 → y = -2x + 6 (slope = -2)
- Since (1/2) * (-2) = -1, these lines are **perpendicular**.

14. **x = 4** and **x = -6**
- These lines are vertical and do not have slopes, thus they are **parallel**.

15. **x = 1** and **y = -8**
- These lines are not vertical or horizontal, hence **neither**.

### Summary of Relationships:
1. Perpendicular
2. Neither
3. Neither
4. Parallel
5. Neither
6. Parallel
7. Neither
8. Perpendicular
9. Parallel
10. Parallel
11. Neither
12. Neither
13. Perpendicular
14. Parallel
15. Neither