https://file.aiquickdraw.com/mathbot/file/1728971679263m3yyursq.jpg

To determine how many solutions this system of equations has, let’s analyze it step-by-step. The system is:

\[
\begin{cases}
y + \frac{2}{3}x = 4 \\
2x = 12 - 3y
\end{cases}
\]

### Step 1: Rewrite Both Equations in Slope-Intercept Form
To see if the lines represented by these equations are parallel, intersecting, or the same line (which tells us about the number of solutions), we can rewrite each equation in the form \( y = mx + b \).

#### First Equation:
\[
y + \frac{2}{3}x = 4
\]
Subtract \(\frac{2}{3}x\) from both sides:
\[
y = -\frac{2}{3}x + 4
\]

This equation has a slope of \(-\frac{2}{3}\) and a y-intercept of 4.

#### Second Equation:
\[
2x = 12 - 3y
\]
To isolate \(y\), first add \(3y\) to both sides:
\[
3y = 12 - 2x
\]
Now, divide by 3:
\[
y = -\frac{2}{3}x + 4
\]

This equation also has a slope of \(-\frac{2}{3}\) and a y-intercept of 4.

### Step 2: Determine the Number of Solutions
Since both equations simplify to the same line (\( y = -\frac{2}{3}x + 4 \)), they are identical. This means every point on one line is also on the other line, and thus, the system has **infinitely many solutions**.

### Conclusion
The system of equations has **infinitely many solutions**.

Would you like additional explanation, or have any questions?

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Here are 5 related questions to consider:
1. What conditions on the slopes and intercepts lead to a system with no solutions?
2. How would you identify if two equations represent parallel lines?
3. What is the graphical interpretation of a system with exactly one solution?
4. How can you verify solutions if given specific values for \(x\) and \(y\)?
5. What is the role of the determinant in determining the number of solutions in a system of linear equations?

**Tip:** When examining systems of equations, rewriting equations in slope-intercept form helps to quickly determine relationships between lines (parallel, identical, or intersecting).

How many solutions does the system of equations have? 
{ y + (2/3)x = 4, 2x = 12 - 3y }

Explore how to determine the number of solutions for a system of linear equations. This problem involves analyzing equations in slope-intercept form, understanding parallel lines, and recognizing identical lines to conclude that the system has infinitely many solutions. Suitable for students in Grades 8-10.

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