1. **State the problem:** We are given the linear function $f(x) = -\frac{1}{3}x + 7$ and need to find the input $x$ that produces an output value of $\frac{2}{3}$. 2. **Write the equation:** Set the function equal to the output value: $$\frac{2}{3} = -\frac{1}{3}x + 7$$ 3. **Isolate the term with $x$:** Subtract 7 from both sides: $$\frac{2}{3} - 7 = -\frac{1}{3}x$$ Convert 7 to thirds: $$\frac{2}{3} - \frac{21}{3} = -\frac{1}{3}x$$ Simplify the left side: $$-\frac{19}{3} = -\frac{1}{3}x$$ 4. **Solve for $x$:** Divide both sides by $-\frac{1}{3}$, which is the same as multiplying by $-3$: $$x = \frac{-\frac{19}{3}}{-\frac{1}{3}} = -\frac{19}{3} \times \cancel{-3} = 19$$ 5. **Interpretation:** The input value $x = 19$ gives the output $f(x) = \frac{2}{3}$. **Final answer:** $$x = 19$$