<s>[INST] Solve the system of equations $\frac{x}{3} + \frac{y}{2} = \frac{2x}{3} - \frac{y}{6} = 7$ for $x$ and $y$. Here is the problem. The answer is suppose to be x = 12, y = 6
$$\frac{ x }{3} + \frac{y}{2} = \frac{ 2x }{3} - \frac{ y }{6} = 7$$
I've tried this. I'm having trouble figuring out what do on the last step I will describe.
$$6 • \frac{ x }{3} + \frac{y}{2} = \frac{ 2x }{3} - \frac{ y }{6} = 7$$
$$2x + 3y = 4x - y = 42$$
$$2x - 4y = 42$$ 
$$4y = 42 - 2x$$
$$y = \frac{ 42 - 2x}{4}$$
$$2x + 3[ \frac{ 42 - 2x }{4} ] = 7$$
 [/INST] You are really given two equations here:
$$
\left\{\begin{align}
  \frac{x}{3} + \frac{y}{2} &= 7\\[0.3cm]
  \frac{2x}{3} - \frac{y}{6} &= 7
\end{align}
\right.
$$
Multiply the first equation by 6 to get $2x + 3y = 42$.  Multiply the second equation by 6 to get $4x - y = 42$.
$$
\left\{
\begin{align} \displaystyle
  2x + 3y &= 42\\
  4x - y &= 42
\end{align}
\right.
$$
Multiply the second equation by $3$ and add it to the first equation to end up with $14x = 168$, and so $x = 12$.  Then use either equation to get $y = 6$.
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