Let's analyze the system of equations given:
1. [tex]\( y = \frac{1}{3} x - 4 \)[/tex]
2. [tex]\( 3y - x = -7 \)[/tex]
To determine the relationships between these lines, we need to consider several properties:
### Step 1: Slope and Y-Intercept
First, we rewrite both equations in slope-intercept form ( [tex]\( y = mx + c \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the y-intercept).
For the first equation:
[tex]\[ y = \frac{1}{3} x - 4 \][/tex]
Here, the slope [tex]\( m \)[/tex] is [tex]\( \frac{1}{3} \)[/tex] and the y-intercept is [tex]\( -4 \)[/tex].
For the second equation, we need to solve for [tex]\( y \)[/tex]:
[tex]\[ 3y - x = -7 \][/tex]
[tex]\[ 3y = x - 7 \][/tex]
[tex]\[ y = \frac{1}{3}x - \frac{7}{3} \][/tex]
Here, the slope [tex]\( m \)[/tex] is [tex]\( \frac{1}{3} \)[/tex], and the y-intercept is [tex]\( -\frac{7}{3} \)[/tex].
### Step 2: Check If They Have the Same Slope
Both lines have the slope [tex]\( \frac{1}{3} \)[/tex]. So, they have the same slope.
### Step 3: Check If They Have the Same Y-Intercept
The y-intercepts are:
- For the first equation: [tex]\( -4 \)[/tex]
- For the second equation: [tex]\( -\frac{7}{3} \)[/tex]
The y-intercepts are different.
### Step 4: Analyze the Relationship Between the Lines
- Parallel Lines: Since both lines have the same slope and different y-intercepts, they are parallel lines.
- Same Line: In order to represent the same line, the lines must have both the same slope and the same y-intercept, which is not the case here.
### Step 5: Check If the System Has One Solution
Since the lines are parallel and have different y-intercepts, they do not intersect. Therefore, the system has no solution.
### Summary of Properties:
1. The system does not have one solution. (False)
2. The system consists of parallel lines. (True)
3. Both lines have the same slope. (True)
4. Both lines have the same y-intercept. (False)
5. The equations do not represent the same line. (False)
### Conclusion
The two correct statements about the system are:
- The system consists of parallel lines.
- Both lines have the same slope.
