Question

Line l1 passes through (-2, 5), and (-1, -10)
Line l2 passes through (5,15), and (3,8).

Find the coordinates of the intersection between the lines. Show all of your work.

need the midpoint for both and then add both midpoints together need step by step soon as possible ty

1. The point of intersection of the two lines is (x, y) = (- 45 / 37, – 250 / 37).

### How to find the point of intersection of two lines by vectorial methods

Vectorially speaking, a line is represented by the following parametric expression:
(x, y) = (a, b) + t · (c – a, d – b)        (1)
Where:
• (a, b) – Initial end of the line segment.
• (c, d) – Final end of the line segment.
• t – Parameter.
Two points intercept each other when they share the same point (x, y). Then,
(- 2, 5) + t₁ · (1, – 15) = (5, 15) + t₂ · (- 2, – 7)
t₁ · (1, – 15) – t₂ · (- 2, – 7) = (7, 10)
t₁ · (1, – 15) + t₂ · (2, 7) = (7, 10)
Which is equivalent to the following system of linear equations:
t₁ + 2 · t₂ = 7
– 15 · t₁ + 7 · t₂ = 10
Then, the solution of the system is (t₁, t₂) = (29 / 37, 115 / 37) and the point of intersection is:
(x, y) = (- 2, 5) + (29 / 37) · (1, – 15)
(x, y) = (- 45 / 37, – 250 / 37)
The point of intersection of the two lines is (x, y) = (- 45 / 37, – 250 / 37).