To determine the solution to the given system of equations, let's analyze the equations step-by-step.
We have the following two linear equations:
1. [tex]\( y = -\frac{1}{3}x + 3 \)[/tex]
2. [tex]\( y = -\frac{1}{3}x - 1 \)[/tex]
### Step-by-Step Solution:
1. Identify the Slope and Y-Intercept:
- Both lines have the same slope of [tex]\( -\frac{1}{3} \)[/tex], as this value is in front of [tex]\( x \)[/tex] in both equations.
- The y-intercept for the first equation is 3 (where the line crosses the y-axis when [tex]\( x = 0 \)[/tex]).
- The y-intercept for the second equation is -1.
2. Graphing the Equations:
- The first line [tex]\( y = -\frac{1}{3}x + 3 \)[/tex] will be a line that starts at [tex]\( (0, 3) \)[/tex] and has a downward slope, moving down 1 unit for every 3 units it moves to the right.
- The second line [tex]\( y = -\frac{1}{3}x - 1 \)[/tex] will be a parallel line that starts at [tex]\( (0, -1) \)[/tex] and follows the same slope, moving down 1 unit for every 3 units it moves to the right.
3. Determine the Relationship between the Lines:
- Since the slopes of both lines are the same ([tex]\( -\frac{1}{3} \)[/tex]), the lines are parallel to each other.
- Parallel lines never intersect. This means there are no points [tex]\((x, y)\)[/tex] that satisfy both equations simultaneously.
### Conclusion:
The given system of equations represents two parallel lines that do not intersect. Therefore, there are no common solutions to the system.
The correct answer is:
[tex]\[ \text{no solution: } \{\} \][/tex]
