MATH OLYMPIAD
\[
\begin{array}{l}
\left\{\begin{array}{l}
6^{x}+6^{y}=42 \\
x+y=3
\end{array}\right. \\
x=? y=?
\end{array}
\]
Answer
Substitute \(x=2\) and \(y=1\) into the original equations, we find this works.
Step 1 ：Let's denote \(6^x=a\) and \(6^y=b\).
Step 2 ：Then, the equations become \(a+b=42\) and \(x+y=3\).
Step 3 ：From the second equation, we can express \(y\) as \(y=3-x\).
Step 4 ：Substitute \(y\) into the first equation, we get \(a+6^{3-x}=42\).
Step 5 ：Rearrange the equation, we get \(a=42-6^{3-x}\).
Step 6 ：Substitute \(a\) into the second equation, we get \(x+\frac{\log(42-6^{3-x})}{\log(6)}=3\).
Step 7 ：Solving this equation, we get \(x=\boxed{2}\) and \(y=\boxed{1}\).
Step 8 ：Substitute \(x=2\) and \(y=1\) into the original equations, we find this works.
