## Simultaneous Equations With Indices Problem

Any set of equations in the same variables are simultaneous equations. This is an example of simultaneous equations with indices.
$100 \times 10^x=10^{2y}$

$\frac{10 \times 10^x}{10^y}=100$

We can write the first of the as
$10^2 \times 10^x=10^{2y} \rightarrow 2+x=2y$

We can write the second of these as
$\frac{10^1 \times 10^x}{10^y}=10^2 \rightarrow 1+x-y=2$

We now have the equations
$2+x=2y$
(1)
$1+x-y=2$
(2)
Thje first of these minus the second gives
$(2+x)-(1+x-y)=2y-2 \rightarrow 1+y=2y-2 \rightarrow 1+2=2y-y \rightarrow y=3$
.
Substiture
$y=3$
into (1) then
$x=2y-2=4$