https://file.aiquickdraw.com/mathbot/file/17273586673946hf1trx0.jpg

The system of equations presented in the image is:

\[
y = 3x - 2
\]
\[
-3x + y = 7
\]

We are asked how many solutions this system has.

### Step 1: Rewrite both equations in slope-intercept form

The first equation is already in slope-intercept form:

\[
y = 3x - 2
\]

The second equation, \(-3x + y = 7\), can be rewritten as:

\[
y = 3x + 7
\]

### Step 2: Compare the equations

We now have:

1. \( y = 3x - 2 \)
2. \( y = 3x + 7 \)

Both lines have the same slope (\(3\)) but different y-intercepts (\(-2\) and \(7\)). Since they are parallel and have different y-intercepts, they never intersect.

### Conclusion:

The system has **0 solutions** because the lines are parallel and do not intersect.

Do you need more details or explanations?

### Related Questions:

1. What is the graphical meaning of parallel lines in a system of equations?
2. How do we classify systems of equations by their number of solutions?
3. What is the general approach to solving a system of linear equations?
4. How can we use substitution to solve a system of equations?
5. What are consistent, inconsistent, and dependent systems?

### Tip:

When solving systems of linear equations, always check if the lines are parallel by comparing their slopes. Parallel lines mean no solutions.

How many solutions does the system of equations y = 3x - 2 and -3x + y = 7 have: 0, 1, or ∞?

Solve the system of linear equations y = 3x - 2 and -3x + y = 7 to find the number of solutions. The problem involves comparing slopes and determining if the lines are parallel, intersecting, or coincident. Suitable for students in Grades 8-10.

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