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

giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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

-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.