Let's solve the given system of equations step by step.
The system is:
[tex]\[
\begin{cases}
\frac{2}{3} x - \frac{1}{3} y = 3 \\
\frac{5}{3} x + \frac{1}{3} y = 4
\end{cases}
\][/tex]
First, let's clear the denominators by multiplying each equation by 3:
[tex]\[
\begin{cases}
3 \left( \frac{2}{3} x - \frac{1}{3} y \right) = 3 \cdot 3 \\
3 \left( \frac{5}{3} x + \frac{1}{3} y \right) = 3 \cdot 4
\end{cases}
\][/tex]
This simplifies to:
[tex]\[
\begin{cases}
2x - y = 9 \\
5x + y = 12
\end{cases}
\][/tex]
Now we have a system of equations in a simpler form:
[tex]\[
\begin{cases}
2x - y = 9 \\
5x + y = 12
\end{cases}
\][/tex]
Next, we add the two equations to eliminate [tex]\( y \)[/tex]:
[tex]\[
(2x - y) + (5x + y) = 9 + 12
\][/tex]
This simplifies to:
[tex]\[
7x = 21
\][/tex]
Solving for [tex]\( x \)[/tex], we get:
[tex]\[
x = \frac{21}{7} = 3
\][/tex]
Now, we substitute [tex]\( x = 3 \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]. Let's use the first equation:
[tex]\[
2x - y = 9
\][/tex]
Substituting [tex]\( x = 3 \)[/tex]:
[tex]\[
2(3) - y = 9
\][/tex]
This simplifies to:
[tex]\[
6 - y = 9
\][/tex]
Solving for [tex]\( y \)[/tex], we get:
[tex]\[
-y = 9 - 6 \\
-y = 3 \\
y = -3
\][/tex]
So, the solution to the system of equations is:
[tex]\[
x = 3, \quad y = -3
\][/tex]
Therefore, the solution is:
[tex]\[
\boxed{(3, -3)}
\][/tex]
