Simultaneous equations to solve unknown exponents Solved the following simultaneous equations for $a$ and $b$:
(1)
$$a + 2b = 2\\3a + b = 10$$
$$3a + 6b = 6\\3a + b = 10$$
$$5b = -4$$
$$b = \frac{-4}{5}\\a = \frac{18}{5}$$
(2)
How do we use the answers/information in (1) to solve the following equations for $x$ and $y$?
$$2^{x+2y} = 2^{20}\\5^{5x+y} = 25^{x+50}$$
In general how do these types of equations relate?

Note that $25^{x+50}=(5^2)^{x+50}=5^{2x+100}$ so the system can be written as
$$2^{x+2y} = 2^{20}\\5^{5x+y} = 5^{2x+100}$$
or
$$2^{x+2y} = 2^{20}\\5^{3x+y} = 5^{100}$$
and comparing exponents we obtain
$$x+2y=20$$
$$3x+y=100$$
This is very similar to the system
$$a + 2b = 2\\3a + b = 10$$
but the RHS is multiplied by $10$. So multiplying by $10$ we have
$$10a + 2(10b) =x+2y= 20\\3(10a) + 10b =3x+y= 100$$
where $x=10a$ and $y=10b$ and you have already found $a$ and $b$.
