Question

Solve the following systems of equations by the method of cross-multiplication:
$\frac{5}{x+y}-\frac{2}{x-y}=-1$
$\frac{15}{x+y}+\frac{7}{x-y}=10$
where $x\ne 0$ , $y\ne 0$ and $x\ne y$

Answer 1

Substitute $\frac{1}{x+y}=a$ and $\frac{1}{x-y}=b$

We get equations as follows: $5a-2b+1=0$ and $15a+7b-10=0$

Using method of cross-multiplication we get:

$\frac{a}{(-2)(-10)-\left(7\right)\left(1\right)}=\frac{b}{\left(1\right)\left(15\right)-(-10)\left(5\right)}=\frac{1}{\left(5\right)\left(7\right)-\left(15\right)(-2)}$

$\frac{a}{13}=\frac{b}{65}=\frac{1}{65}$

From $\frac{a}{13}=\frac{1}{65}$ we get $a=\frac{1}{5}$

Similarly $b=1$

Resubstituting values of $a$ and $b$ we get $x+y=5$ and $x-y=1$

Adding these 2 equations gives us $x=3$ .

Substracting 2nd equation from 1st gives us $y=2$