For the line given by the equation [tex]y = 0.5x + 4[/tex], determine if it intersects or is parallel to the line [tex]y = -1.5x[/tex]. If it intersects the line, find the intersection point.



Answer :

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].