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:
-
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.
-
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.
-
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.
-
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.
-
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.
-
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.
-
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.
-
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.
-
-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.
-
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.
-
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.
-
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.
-
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.
-
x = 4 and x = -6
- These lines are vertical and do not have slopes, thus they are parallel.
-
x = 1 and y = -8
- These lines are not vertical or horizontal, hence neither.
Summary of Relationships:
- Perpendicular
- Neither
- Neither
- Parallel
- Neither
- Parallel
- Neither
- Perpendicular
- Parallel
- Parallel
- Neither
- Neither
- Perpendicular
- Parallel
- Neither