What is the equation of a line passing through the intersection of two other lines and a point?L1 y=-x/2 L2 y=2x+5 point (1,2)

We have to find the equation of the line that is passing
through two points. One point is given. The next point represents the intercepting point
of the given lines. To determine this point, we'll have to solve the system of the
equations of the lines.

y=-x/2 - 5
(1)

y=2x+5 (2)

We'll equate
(1) and (2):

-x/2 - 5 = 2x +
5

We'll subtract 2x both
sides:

-x/2 - 2x - 5 = 5

We'll
add 5 both sides:

-5x/2 =
5+5

-5x/2 = 10

-5x = 20
=> x = -4

We'll replace x in
(2):

y = -8 + 5

y =
-3

The intercepting point of the lines L1 and L2 is: (-4 ;
-3).

Now, we'll write the equation of the line that passes
through the points (1 ; 2) and (-4 ; -3).

(x2 - x1)/(x -
x1) = (y2 - y1)/(y - y1)

(-4 - 1)/(x - 1) = (-3 - 2)/(y -
2)

-5/(x - 1) = -5/(y - 2)

x -
1 = y - 2

The equation of the line that
passes through the point (1 ; 2) and the intercepting point of the lines L1 and L2 is: x
- y + 1 = 0.