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

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### 1 Answer

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)))**