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.

Answer 1

To determine the relationship between line BC (represented by the equation \(3x + 2y = 8\)) and line AD (represented by the equation \(-3x - 2y = 6\)), we can examine both equations.

First, let's rewrite both equations in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

### For line BC:
Starting from the equation:
\[
3x + 2y = 8
\]
We can isolate \(y\):
\[
2y = -3x + 8
\]
\[
y = -\frac{3}{2}x + 4
\]
So, the slope (\(m\)) of line BC is \(-\frac{3}{2}\).

### For line AD:
Starting from the equation:
\[
-3x - 2y = 6
\]
We can isolate \(y\):
\[
-2y = 3x + 6
\]
Dividing by -2:
\[
y = -\frac{3}{2}x - 3
\]
So, the slope (\(m\)) of line AD is \(-\frac{3}{2}\).

### Relationship between lines:
Since both lines have the same slope of \(-\frac{3}{2}\), they are parallel to each other.

### Examining the sum of the equations:
Now let's analyze the sum of the two equations:

1. **Equation of line BC**: \(3x + 2y = 8\)
2. **Equation of line AD**: \(-3x - 2y = 6\)

Adding both equations:
\[
(3x + 2y) + (-3x - 2y) = 8 + 6
\]
This simplifies to:
\[
0 = 14
\]

The left side equals zero, which shows that the two lines do not intersect, confirming that they are parallel (since parallel lines do not intersect at any point).

### Conclusion:
The relationship between line BC and line AD is that they are parallel lines. The sum of their equations resulting in a false statement (\(0 = 14\)) illustrates that there is no solution simultaneous to both equations, reinforcing the fact that the lines do not intersect.