Question
Line BC is represented by 3x + 2y = 8. Line AD is represented by –3x – 2y = 6. What is the relationship of line BC to line AD? Explain how the sum of the equations demonstrates this relationship. make the writing short
Answers
Answered by
GPT-4o mini
To determine the relationship between lines BC and AD, we can rewrite their equations in slope-intercept form (y = mx + b).
1. For line BC, \(3x + 2y = 8\):
\[
2y = -3x + 8 \quad \Rightarrow \quad y = -\frac{3}{2}x + 4
\]
(slope = -3/2)
2. For line AD, \(-3x - 2y = 6\) can be rearranged as:
\[
-2y = 3x + 6 \quad \Rightarrow \quad y = -\frac{3}{2}x - 3
\]
(slope = -3/2)
Both lines have the same slope of -3/2, indicating they are parallel.
To confirm this using their equations, if we sum them:
\[
(3x + 2y) + (-3x - 2y) = 8 + 6 \quad \Rightarrow \quad 0 = 14
\]
This contradiction shows that the lines do not intersect, reinforcing that they are parallel.
1. For line BC, \(3x + 2y = 8\):
\[
2y = -3x + 8 \quad \Rightarrow \quad y = -\frac{3}{2}x + 4
\]
(slope = -3/2)
2. For line AD, \(-3x - 2y = 6\) can be rearranged as:
\[
-2y = 3x + 6 \quad \Rightarrow \quad y = -\frac{3}{2}x - 3
\]
(slope = -3/2)
Both lines have the same slope of -3/2, indicating they are parallel.
To confirm this using their equations, if we sum them:
\[
(3x + 2y) + (-3x - 2y) = 8 + 6 \quad \Rightarrow \quad 0 = 14
\]
This contradiction shows that the lines do not intersect, reinforcing that they are parallel.