Equation Alignment for Table Mapping of X and Y

Which of the following equations corresponds to the function represented in the table?

To determine which equation corresponds to the function given by the table, we will test each equation using the $(x, y)$ pairs from the table. Specifically, we will verify which equation satisfies all pairs so that we can conclude it functions as desired.

Consider the equation $y = \frac{2}{3}x + 2\frac{2}{3}$.

• For $x = -1$:

Substitute into the equation:
$y = \frac{2}{3}(-1) + 2\frac{2}{3} = -\frac{2}{3} + \frac{8}{3} = \frac{6}{3} = 2$. The value matches the table, $y = 2$.

• For $x = 2$:

Substitute into the equation:
$y = \frac{2}{3}(2) + 2\frac{2}{3} = \frac{4}{3} + \frac{8}{3} = \frac{12}{3} = 4$. The value matches the table, $y = 4$.

• For $x = 5$:

Substitute into the equation:
$y = \frac{2}{3}(5) + 2\frac{2}{3} = \frac{10}{3} + \frac{8}{3} = \frac{18}{3} = 6$. The value matches the table, $y = 6$.

• For $x = 8$:

Substitute into the equation:
$y = \frac{2}{3}(8) + 2\frac{2}{3} = \frac{16}{3} + \frac{8}{3} = \frac{24}{3} = 8$. The value matches the table, $y = 8$.

• For $x = 11$:

Substitute into the equation:
$y = \frac{2}{3}(11) + 2\frac{2}{3} = \frac{22}{3} + \frac{8}{3} = \frac{30}{3} = 10$. The value matches the table, $y = 10$.

Thus, the equation $y = \frac{2}{3}x + 2\frac{2}{3}$ satisfies all pairs from the table, confirming it is the correct representation.

Therefore, the correct answer is $y = \frac{2}{3}x + 2\frac{2}{3}$.