1. **Stating the problem:** We are given a table with values of $z$ and the corresponding values of the expression $\frac{2z}{3} + 1$. We want to understand how to compute $\frac{2z}{3} + 1$ for given $z$ values and interpret the results. 2. **Formula used:** The formula to compute the second column is: $$y = \frac{2z}{3} + 1$$ where $z$ is the input value and $y$ is the output. 3. **Calculations for given $z$ values:** - For $z=1$: $$y = \frac{2 \times 1}{3} + 1 = \frac{2}{3} + 1 = \frac{2}{3} + \frac{3}{3} = \frac{5}{3} = 1 \frac{2}{3}$$ - For $z=2$: $$y = \frac{2 \times 2}{3} + 1 = \frac{4}{3} + 1 = \frac{4}{3} + \frac{3}{3} = \frac{7}{3} = 2 \frac{1}{3}$$ - For $z=-5$: $$y = \frac{2 \times (-5)}{3} + 1 = \frac{-10}{3} + 1 = \frac{-10}{3} + \frac{3}{3} = \frac{-7}{3} = -2 \frac{1}{3}$$ 4. **Explanation:** To find the value of $y$ for any $z$, multiply $z$ by $\frac{2}{3}$ and then add 1. The results can be expressed as improper fractions or mixed numbers for easier interpretation. 5. **Summary:** - $z=1$ gives $y=1 \frac{2}{3}$ - $z=2$ gives $y=2 \frac{1}{3}$ - $z=-5$ gives $y=-2 \frac{1}{3}$ This matches the values in the table and shows how the formula transforms $z$ values. **Final answer:** $$y = \frac{2z}{3} + 1$$