Question

the point P(x,y) lies on the straight line joining A(3,0) and B(5,6). find expressions for the gradients of AP and PB hence show that y=3x-9.
explain in steps

Answer 1

A(3 ,0)                            P(x ,y)                    B(5, 6)
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Gradient = Δy/Δx
1st Grad = (y-0)/(x-3) = y/(x-3)
2nd Grad = (y-6)/(x-5)
Since this is a straight line, the gradient is uniform i.e. 1st Grad = 2nd Grad

⇒y/(x-3) = (y-6)/(x-5)
(y-6)(x-3) = y(x-5)
xy-3y-6x+18 = xy-5y
xy-xy-3y+5y = 6x-18
         2y = 6x-18
⇒ y = 3x-9 (Proved)