Solve the System of Equations: 2x/3 - y = 3 and x + 3y = 6

$\frac{2x}{3}-y=3$

$x+3y=6$

To solve this system of equations, we'll use the substitution method.

Here are the steps we will take:

• Step 1: Solve the second equation $x + 3y = 6$ for $x$.
• Step 2: Substitute the expression for $x$ in the first equation $\frac{2x}{3} - y = 3$.
• Step 3: Solve the resulting equation for $y$.
• Step 4: Substitute back to find $x$.

Let's begin:

Step 1: Solve $x + 3y = 6$ for $x$:
$x = 6 - 3y$

Step 2: Substitute $x = 6 - 3y$ into the first equation $\frac{2x}{3} - y = 3$:
$\frac{2(6 - 3y)}{3} - y = 3$

Simplify the expression:
$\frac{12 - 6y}{3} - y = 3$

This simplifies to:
$4 - 2y - y = 3$

Combine like terms:
$4 - 3y = 3$

Isolate $y$:
$-3y = 3 - 4$
$-3y = -1$
$y = \frac{-1}{-3}$
$y = \frac{1}{3}$

Step 3: With $y = \frac{1}{3}$, substitute back into $x = 6 - 3y$:
$x = 6 - 3\left(\frac{1}{3}\right)$
$x = 6 - 1$
$x = 5$

Therefore, the solution to this system of equations is $\mathbf{x = 5, y = \frac{1}{3}}$.

Referring to the choice list, the correct choice is Choice 3: $x = 5, y = \frac{1}{3}$.