To determine how many solutions the system of equations has, we can analyze the given equations:
-
First Equation: \( y = 3x - 2 \)
- This is in slope-intercept form \( y = mx + b \), where the slope \( m = 3 \) and the y-intercept \( b = -2 \).
-
Second Equation: \( -3x + y = 7 \)
- We can rearrange this equation into slope-intercept form: \[ y = 3x + 7 \]
- This has a slope \( m = 3 \) and a y-intercept \( b = 7 \).
Now let's analyze the slopes and y-intercepts:
- Both lines have the same slope of \( 3 \).
- The y-intercepts are different (\(-2\) for the first equation and \(7\) for the second equation).
Since both lines are parallel (have the same slope) but have different y-intercepts, they will never intersect.
Thus, the system of equations has 0 solutions.