Math, asked by shivangbmehta7852, 9 months ago

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

Answers

Answered by MaheswariS
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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