Question

Solve the equations:
$\frac{(m+n)x-(a-b)}{(m-n)x-(a+b)}=\frac{(m+n)x+a+c}{(m-n)x+a-c}$.

Answer 1

Use $\frac{x}{y}=\frac{x+y}{x-y}$

Apply this property on both sides of the expression

$\frac{(m+n)x-(a-b)}{(m-n)x-(a+b)}=\frac{(m+n)x+a+c}{(m-n)x+a-c}$

$\Rightarrow \frac{2mx-2a}{2nx+2b}=\frac{2mx+2a}{2nx+2c}$

$\Rightarrow \frac{mx-a}{nx+b}=\frac{mx+a}{nx+c}$

Now cross multiply and simplify the equation

$\Rightarrow (mx-a)(nx+c)=(nx+b)(mx+a)$

$\Rightarrow mn{x}^2+x(mc-an)-ac=mn{x}^2+x(an+bm)+ab$

$\Rightarrow x\left(m\right(c-b)-2an)=a(b+c)$

$\Rightarrow x=\frac{a(b+c)}{m(c-b)-2an}$