## Simultaneous Equations in Powers of x and y

We can solve the simultaneous equations
$x^3y_2=72$
(1)
$x^2y^4=324$
(2)
May making the powers of either
$x$
or
$y$
the same, then dividing one equation by the other, which cancels either
$x$
or
$y$
.
We can make the coefficients of
$y$
the same by squaring equation )1_. We obtain
$x^6y_4=72$
(3)
(3) divided by (2) gives
$\frac{x^6y_4}{x^2y^4}=\frac{72^2}{324} x^4=15 \rightarrow x=-2, \: 2$

In fact
$x$
cannot be equal to -2 because from the first equation
$x^3 = \frac{72}{y^2} \gt 0$
.
Then from (1),
$y=\sqrt{\frac{72}{x^3}}=\sqrt{\frac{72}{2^3}}=-3, \: 3$
.
$x=2, \: y=-3$
and
$x=2, \: y=3$
satisfy both equations so are the solutions.