Question

Solve each of the following systems of equations by the method of cross-multiplication:
ax + by = a²
bx + ay = b²

Answer 1

$\text{Given equations are}$

$ax+by=a^2$

$bx+ay=b^2$

$\text{It can be written as}$

$\bf\,ax+by-a^2=0$

$\bf\,bx+ay-b^2=0$

$\text{By cross multiplication rule,}$

$\displaystyle\frac{x}{-b^3+a^3}=\frac{y}{-a^2b+ab^2}=\frac{1}{a^2-b^2}$

$\displaystyle\frac{x}{a^3-b^3}=\frac{y}{ab(b-a)}=\frac{1}{a^2-b^2}$

$\displaystyle\frac{x}{(a-b)(a^2+ab+b^2)}=\frac{y}{-ab(a-b)}=\frac{1}{(a-b)(a+b)}$

$\displaystyle\frac{x}{a^2+ab+b^2}=\frac{y}{-ab}=\frac{1}{a+b}$

$\implies\displaystyle\,x=\frac{a^2+ab+b^2}{a+b}$ $\text{and}$

$\displaystyle\,y=\frac{-ab}{a+b}$

$\therefore\textbf{The solution is }\bf\,x=\frac{a^2+ab+b^2}{a+b},\;y=\frac{-ab}{a+b}$

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2.Solve each of the following systems of equations by the method of cross-multiplication:

(x/a) + (y/b) = a + b

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