Answer :
To determine which of the given points lies on the graph of the equation \( y = \frac{1}{3}x + 2 \), I'll test each point by substituting the \( x \) and \( y \) coordinates into the equation and checking if the equality holds.
### Step-by-Step Solution:
1. Point A: \( (-3, 2) \)
Substitute \( x = -3 \) and \( y = 2 \) into the equation:
[tex]\[ y = \frac{1}{3}(-3) + 2 \][/tex]
Simplify:
[tex]\[ y = -1 + 2 = 1 \][/tex]
Since \( 2 \neq 1 \), point \( (-3, 2) \) does not lie on the graph.
2. Point B: \( (3, 5) \)
Substitute \( x = 3 \) and \( y = 5 \) into the equation:
[tex]\[ y = \frac{1}{3}(3) + 2 \][/tex]
Simplify:
[tex]\[ y = 1 + 2 = 3 \][/tex]
Since \( 5 \neq 3 \), point \( (3, 5) \) does not lie on the graph.
3. Point C: \( (3, 3) \)
Substitute \( x = 3 \) and \( y = 3 \) into the equation:
[tex]\[ y = \frac{1}{3}(3) + 2 \][/tex]
Simplify:
[tex]\[ y = 1 + 2 = 3 \][/tex]
Since \( 3 = 3 \), point \( (3, 3) \) does lie on the graph.
4. Point D: \( (6, 8) \)
Substitute \( x = 6 \) and \( y = 8 \) into the equation:
[tex]\[ y = \frac{1}{3}(6) + 2 \][/tex]
Simplify:
[tex]\[ y = 2 + 2 = 4 \][/tex]
Since \( 8 \neq 4 \), point \( (6, 8) \) does not lie on the graph.
### Conclusion:
From the evaluations above, the point that lies on the graph of the equation \( y = \frac{1}{3}x + 2 \) is:
C) [tex]\( (3, 3) \)[/tex].
### Step-by-Step Solution:
1. Point A: \( (-3, 2) \)
Substitute \( x = -3 \) and \( y = 2 \) into the equation:
[tex]\[ y = \frac{1}{3}(-3) + 2 \][/tex]
Simplify:
[tex]\[ y = -1 + 2 = 1 \][/tex]
Since \( 2 \neq 1 \), point \( (-3, 2) \) does not lie on the graph.
2. Point B: \( (3, 5) \)
Substitute \( x = 3 \) and \( y = 5 \) into the equation:
[tex]\[ y = \frac{1}{3}(3) + 2 \][/tex]
Simplify:
[tex]\[ y = 1 + 2 = 3 \][/tex]
Since \( 5 \neq 3 \), point \( (3, 5) \) does not lie on the graph.
3. Point C: \( (3, 3) \)
Substitute \( x = 3 \) and \( y = 3 \) into the equation:
[tex]\[ y = \frac{1}{3}(3) + 2 \][/tex]
Simplify:
[tex]\[ y = 1 + 2 = 3 \][/tex]
Since \( 3 = 3 \), point \( (3, 3) \) does lie on the graph.
4. Point D: \( (6, 8) \)
Substitute \( x = 6 \) and \( y = 8 \) into the equation:
[tex]\[ y = \frac{1}{3}(6) + 2 \][/tex]
Simplify:
[tex]\[ y = 2 + 2 = 4 \][/tex]
Since \( 8 \neq 4 \), point \( (6, 8) \) does not lie on the graph.
### Conclusion:
From the evaluations above, the point that lies on the graph of the equation \( y = \frac{1}{3}x + 2 \) is:
C) [tex]\( (3, 3) \)[/tex].
