Equation Alignment for Table Mapping of X and Y

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