Question

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

$\frac{x}{a}+\frac{y}{b}=2$

$ax-by=c$

Answer 1

Given equations:-

$\frac{x}{a}+\frac{y}{b}=2=>bx+ay-2ab=0ax-by=c=>ax-by-c=0$

Using cross-multiplication method,

$\frac{x}{\left(a\right)(-c)-(-b)(-2ab)}=\frac{y}{(-2ab)\left(a\right)-(-c)\left(b\right)}=\frac{1}{\left(b\right)(-b)-\left(a\right)\left(a\right)}\frac{x}{-ac-2a{b}^2}=\frac{y}{-2a^{2}b+bc}=\frac{1}{-b^2-a^2}\frac{x}{-a(c+2b^2)}=\frac{y}{b(c-2a^2)}=\frac{1}{-(a^2+b^2)}=>\frac{x}{-a(c+2b^2)}=\frac{1}{-(a^2+b^2)}or,x=\frac{a(c+2b^2)}{(a^2+b^2)}=>\frac{y}{b(c-2a^2)}=\frac{1}{-(a^2+b^2)}or,y=\frac{b(2a^2-c)}{(a^2+b^2)}\therefore x=\frac{a(c+2b^2)}{(a^2+b^2)}andy=\frac{b(2a^2-c)}{(a^2+b^2)}$