Which of the following points lies on the graph of this equation?

[tex]\[ y = \frac{1}{3}x + 2 \][/tex]

A) \((-3, 2)\)
B) \((3, 5)\)
C) \((3, 3)\)
D) [tex]\((6, 8)\)[/tex]



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].