# What is the point of intersection of the lines ax + by + c = 0 and px + qy + r = 0 in terms of the given constants?

justaguide | College Teacher | (Level 2) Distinguished Educator

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We have to find the point of intersection of the lines ax + by + c = 0 and px + qy + r = 0.

This is done by solving the following set of equations for x and y:

ax + by + c = 0...(1)

px + qy + r = 0....(2)

Multiply (1) by p and (2) by a and subtract the two equations

apx + bpy + cp - apx - aqy - ar = 0

=> bpy - aqy = ar - cp

=> y(bp - aq) = ar - cp

=> y = (ar - cp)/(bp - aq)

substitute this value of y in (1)

ax + b(ar - cp)/(bp - aq) + c = 0

=> ax = -c - b(ar - cp)/(bp - aq)

=> x = -c/a - (b/a)(ar - cp)/(bp - aq)

The solution derived for x and y can be used to find the point of intersection of any two lines. It only requires the substitution of the numeric values of the constants.

The point of intersection of the given lines is: ((-c/a - (b/a)(ar - cp)/(bp - aq), ((ar - cp)/(bp - aq)))