18. The numbers $x$ and $y$ are such that $3^{x}+3^{y+1}=5 \sqrt{3}$ and $3^{x+1}+3^{y}=3 \sqrt{3}$. What is the value of $3^{x}+3^{y}$ ?
A $\sqrt{3}$
B $2 \sqrt{3}$
C $3 \sqrt{3}$
D $4 \sqrt{3}$
E $5 \sqrt{3}$
Solution
B
Note that $3^{x+1}=3\left(3^{x}\right)$ and $3^{y+1}=3\left(3^{y}\right)$.
Therefore the equations given in the question may be rewritten as
$$
3^{x}+3\left(3^{y}\right)=5 \sqrt{3}
$$
and
$$
3\left(3^{x}\right)+3^{y}=3 \sqrt{3} .
$$

Adding equations (1) and (2) gives
$$
4\left(3^{x}+3^{y}\right)=8 \sqrt{3} .
$$

Therefore
$$
3^{x}+3^{y}=2 \sqrt{3} .
$$

FOR INVESTIGATION
18.1 (a) Use equations (1) and (2) to find the values of $3^{x}$ and $3^{y}$.
(b) Check that the values for $3^{x}$ and $3^{y}$ that you have found satisfy equation (4).
18.2 (For those who know about logarithms.)
(a) Use your answer to Problem 18.1 (a) to show that $x=\frac{1}{2}-\frac{\ln 2}{\ln 3}$.
(b) Find a similar expression for the value of $y$.
18.3 Find the values of $x$ and $y$ that satisfy both of the equations
$$
4^{x+1}+5^{y}=281
$$
and
$$
4^{x}+5^{y+1}=189
$$