Question

which of the following linear equations corresponds to the table above?

a. \\(y = \frac{1}{3}x + 2\\)
b. \\(y = \frac{1}{3}x - 2\\)
c. \\(y = 3x + 2\\)
d. \\(y = 3x - 2\\)

Explanation:

Identify the y-intercept

The y-intercept is the value of \(y\) when \(x = 0\). Looking at the table, when \(x = 0\), \(y = 2\). Thus, the y-intercept \(b\) is \(2\).

Calculate the slope

Using the points \((0, 2)\) and \((2, 8)\), calculate the slope \(m\):

$$ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 2}{2 - 0} = \frac{6}{2} = 3 $$

Verify with the third point

Check if the slope holds true for the point \((6, 20)\) using the equation \(y = 3x + 2\):

$$ y = 3(6) + 2 = 18 + 2 = 20 $$

The point satisfies the equation.

Match with options

The linear equation is \(y = 3x + 2\), which corresponds to option C.

Answer:

• (A) \(y = \frac{1}{3}x + 2\)
• (B) \(y = \frac{1}{3}x - 2\)
• (C) \(y = 3x + 2\) (Correct answer)
• (D) \(y = 3x - 2\)