To determine if there are points of intersection for the lines defined by the systems of equations, we can analyze and graph the two given lines.
### Step-by-Step Solution:
1. Identify the Line Equations:
[tex]\[
y = \frac{1}{3}x + 6
\][/tex]
[tex]\[
y = \frac{1}{3}x - 2
\][/tex]
2. Compare the Slopes and Y-Intercepts:
Both lines have the same slope of [tex]\(\frac{1}{3}\)[/tex], which indicates that they are parallel lines because their slopes are equal. However, their y-intercepts are different:
- For the first line [tex]\(\left(y = \frac{1}{3}x + 6\right)\)[/tex], the y-intercept is 6.
- For the second line [tex]\(\left(y = \frac{1}{3}x - 2\right)\)[/tex], the y-intercept is -2.
3. Plot the Lines:
- First Line ([tex]\(y = \frac{1}{3}x + 6\)[/tex]):
- When [tex]\(x = 0\)[/tex], [tex]\(y = 6\)[/tex] (y-intercept).
- When [tex]\(x = 3\)[/tex], [tex]\(y = 7\)[/tex] (since [tex]\((1/3)(3) + 6 = 7\)[/tex]).
- Second Line ([tex]\(y = \frac{1}{3}x - 2\)[/tex]):
- When [tex]\(x = 0\)[/tex], [tex]\(y = -2\)[/tex] (y-intercept).
- When [tex]\(x = 3\)[/tex], [tex]\(y = -1\)[/tex] (since [tex]\((1/3)(3) - 2 = -1\)[/tex]).
4. Conclusion:
Since the slopes are the same and the y-intercepts are different, these lines are parallel and will never intersect.
### Answer:
There are no solutions since the lines are parallel and do not intersect at any point. Therefore, the correct answer is:
[tex]\[
\text{no solution: } \{\}
\][/tex]
