The problem asks us to determine the number of solutions for the given system of equations: $y + \frac{2}{3}x = 4$ $2x = 12 - 3y$

1. Problem Description

The problem asks us to determine the number of solutions for the given system of equations:

$y + \frac{2}{3}x = 4$

$2x = 12 - 3y$

2. Solution Steps

First, let's rewrite the first equation to isolate $y$:

$y = 4 - \frac{2}{3}x$

Now, let's rewrite the second equation to isolate $y$:

$2x = 12 - 3y$

$3y = 12 - 2x$

$y = \frac{12 - 2x}{3}$

$y = 4 - \frac{2}{3}x$

Notice that both equations, when solved for $y$, are identical: $y = 4 - \frac{2}{3}x$. This means that the two equations represent the same line. Since the lines are identical, there are infinitely many points that satisfy both equations.

3. Final Answer

The system of equations has infinitely many solutions because the two equations represent the same line.