## Simultaneous Equations With Indices

The simultaneous equations
$x^2y^3=1125$
(1)
$x^3y^2-675$
(2)
Can most easily be solved by raising each equation to some power so that the powers of
$x$
( or
$y$
are equalised, then dividing the equations
Raise (1) to the power of 3 and (2) to the power of 2 giving
$x^6y^9=1125^3$
(3)
$x^6y^4=675^2$
(4)
(3) divided by (4) gives
$\frac{x^6y^9}{x^6y^4}=\frac{1125^3}{675^2} \rightarrow y^5=3125 \rightarrow y=\sqrt[5]{3125}=5$

Then from (1)
$x^25^3=1125 \rightarrow x^2=\frac{1125}{5^3}=9 \rightarrow x=\sqrt{9}=3$

Only one square root is taken since
$x=-3$
does not fit equation (2)