Question

Solve each of the following systems of equations by using the method of cross multiplication:

$\frac{1}{x}+\frac{1}{y}=7,\frac{2}{x}+\frac{3}{y}=17(x\ne 0andy\ne 0).$

Answer 1

$\frac{1}{x}+\frac{1}{y}=7$ -----(1)

$\frac{2}{x}+\frac{3}{y}=17$ ------(2)

let $\frac{1}{x}=u$ and $\frac{1}{y}=v$

so equation are written as

$u+v-7=0$.....(3)

$2u+3v-17=0$....(4)

by cross multiplication

$\frac{u}{-17+21}=\frac{v}{-14+17}=\frac{1}{3-2}$

$\frac{u}{4}=\frac{v}{3}=\frac{1}{1}$

$\frac{u}{4}=\frac{1}{1}\Rightarrow u=4$
and
$\frac{v}{3}=\frac{1}{1}\Rightarrow v=3$

But $\frac{1}{x}=u$

$\frac{1}{x}=4$

so $x=\frac{1}{4}$

and

$\frac{1}{y}=v$

$\frac{1}{y}=3$

so $y=\frac{1}{3}$

$x=\frac{1}{4}$ and $y=\frac{1}{3}$