Question

Compare equations 2x+3y=6
and 3x−2y=6
. Select the true statement.(1 point)
Responses

The lines are parallel.
The lines are parallel.

The lines are perpendicular.
The lines are perpendicular.

The lines do not intersect.
The lines do not intersect.

The lines intersect at one point: the y
-intercept.

Answer 1

To determine the relationship between the two lines given by the equations \(2x + 3y = 6\) and \(3x - 2y = 6\), we can first rewrite both equations in slope-intercept form (y = mx + b), where m is the slope.

1. **For the first equation \(2x + 3y = 6\)**:
\[
3y = -2x + 6 \\
y = -\frac{2}{3}x + 2
\]
Here, the slope (m) is \(-\frac{2}{3}\).

2. **For the second equation \(3x - 2y = 6\)**:
\[
-2y = -3x + 6 \\
2y = 3x - 6 \\
y = \frac{3}{2}x - 3
\]
Here, the slope (m) is \(\frac{3}{2}\).

Next, we can compare the slopes:
- The slope of the first line is \(-\frac{2}{3}\).
- The slope of the second line is \(\frac{3}{2}\).

Lines are:
- **Parallel** if they have the same slope (not the case here).
- **Perpendicular** if the product of their slopes is \(-1\).

Calculating the product of the slopes:
\[
-\frac{2}{3} \times \frac{3}{2} = -1
\]
Since the product is \(-1\), the lines are indeed **perpendicular**.

**Conclusion**: The true statement is:
- **The lines are perpendicular.**