Encyclopaedia Britannica, 11th Edition, Volume 9, Slice 7

1. Equations which may be reduced to linear equations.--Ex. To solve x(x - a) = yz, y(y - b) = zx, z(z - c)=xy. Multiply the equations by y, z and x respectively, and divide the sum by xyz; then

a b c -- + -- + -- = 0 ... (1). z x y

Multiply by z, x and y, and divide the sum by xyz; then

a b c -- + -- + -- = 0 ... (2). y z x

From (1) and (2) by cross multiplication we obtain

1 1 1 1 ----------- = ----------- = ----------- = -------- (suppose)(3). y(b^2 - ac) z(c^2 - ab) x(a^2 - bc) [lambda]

Substituting for x, y and z in x(x - a) = yz we obtain

1 3abc - (a^3 + b^3 + c^3) -------- = ------------------------------; [lambda] (a^2 - bc)(b^2 - ac)(c^2 - ab)

and therefore x, y and z are known from (3). The same ar