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