Question

Solve the following pair of simultaneous equations for x and y. qx+ (1− p)y = R and px+ (1−q)y = S.

Answer 1

qx + ( 1-p) y = R

px + (1-q)y = S

multiplying equation 1 by -p and equation 2 by q

-pqx - p ( 1- p)y = Rp

pqx + q ( 1 - q) y = Sq

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-p ( 1 - p) y + q( 1- q) y = Rp + Sq

-py + p^2y + qy - q^2 y = Rp + Sq

y ( p^2 - q^2 + q - p ) = Rp + Sq

y = (Rp + Sq) /  ( p^2 - q^2 + q - p )

plugging this value of y into equation 1

qx + ( 1- p ) (Rp + Sq) /  ( p^2 - q^2 + q - p ) = R

qx = R -  ( 1- p ) (Rp + Sq) /  ( p^2 - q^2 + q - p )

x = ( 2Rp^2 - Rq^2 + Rq - 2Rp - Sq + Spq ) / q ( p^2 - q^2 + q - p )