https://file.aiquickdraw.com/mathbot/file/1729708688768y2ni3psj.jpg

The system of linear equations provided in the image is:

\[
\begin{aligned}
-12x + 4y &= 16 \\
-9x + 3y &= 12
\end{aligned}
\]

We want to determine how many solutions this system has. To do this, we can solve it using various methods such as substitution, elimination, or graphically analyzing the slopes and intercepts.

First, let's rewrite both equations in slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

1. **First equation**:
   \[
   -12x + 4y = 16
   \]
   Solving for \( y \):
   \[
   4y = 12x + 16
   \]
   \[
   y = 3x + 4
   \]

2. **Second equation**:
   \[
   -9x + 3y = 12
   \]
   Solving for \( y \):
   \[
   3y = 9x + 12
   \]
   \[
   y = 3x + 4
   \]

Both equations simplify to \( y = 3x + 4 \), which means they are the same line. Since they are identical, the system has **infinitely many solutions**, as both lines overlap completely.

Therefore, the correct answer is **"Infinite"**.

Would you like further details or explanations on this solution?

Here are 5 follow-up questions related to this topic:
1. How can you identify if a system of equations has no solutions by graphing?
2. What does it mean when two linear equations have exactly one solution?
3. How can you solve a system of equations using the substitution method?
4. How do you apply the elimination method to a system of equations?
5. What happens if two lines in a system have the same slope but different y-intercepts?

**Tip:** Whenever you find the same slope and intercept for two equations, the lines are identical, meaning infinite solutions!

How many solutions does this system have? -12x + 4y = 16, -9x + 3y = 12

This problem involves solving a system of linear equations graphically: -12x + 4y = 16 and -9x + 3y = 12. By converting the equations into slope-intercept form, we discover that both equations represent the same line, leading to infinitely many solutions. Ideal for students in Grades 8-10.

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