To determine if the given lines intersect or are parallel, we'll follow these steps:
1. Identify the slopes of both lines:
- For the first line, [tex]\( y = -1.5x \)[/tex], the slope (m1) is [tex]\(-1.5\)[/tex].
- For the second line, [tex]\( y = 0.5x + 4 \)[/tex], the slope (m2) is [tex]\(0.5\)[/tex].
2. Compare the slopes:
- If the slopes are equal, the lines are parallel and do not intersect.
- If the slopes are different, the lines intersect at some point.
3. Calculate the intersection point:
- Since the slopes [tex]\(-1.5\)[/tex] and [tex]\(0.5\)[/tex] are different, the lines intersect.
- To find the intersection point, set the equations of the lines equal to each other:
[tex]\[
-1.5x = 0.5x + 4
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
- Combine like terms:
[tex]\[
-1.5x - 0.5x = 4
\][/tex]
[tex]\[
-2x = 4
\][/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{4}{-2} = -2
\][/tex]
5. Find the corresponding [tex]\( y \)[/tex]-coordinate:
- Substitute [tex]\( x = -2 \)[/tex] back into either line equation. Using [tex]\( y = 0.5x + 4 \)[/tex]:
[tex]\[
y = 0.5(-2) + 4 = -1 + 4 = 3
\][/tex]
6. Conclusion:
- The lines intersect at the point [tex]\( (-2, 3) \)[/tex].
Hence, the two lines intersect at the point [tex]\( (-2, 3) \)[/tex].
