1. **State the problem:** We need to find the rule for the table where $X$ is the input and $Y$ is the output: $$\begin{array}{c|c} X & Y \\ \hline 1 & 5 \\ 3 & 15 \\ 4 & 20 \\ 7 & 35 \\ & 75 \end{array}$$ 2. **Look for a pattern or relationship:** Check if $Y$ is a multiple of $X$. 3. **Calculate ratios:** - For $X=1$, $Y=5$, so $\frac{Y}{X} = \frac{5}{1} = 5$ - For $X=3$, $Y=15$, so $\frac{Y}{X} = \frac{15}{3} = 5$ - For $X=4$, $Y=20$, so $\frac{Y}{X} = \frac{20}{4} = 5$ - For $X=7$, $Y=35$, so $\frac{Y}{X} = \frac{35}{7} = 5$ 4. **Identify the rule:** Since $\frac{Y}{X} = 5$ for all given points, the rule is: $$Y = 5X$$ 5. **Find the missing output for $Y$ when $X$ is unknown but $Y=75$:** $$75 = 5X$$ Divide both sides by 5: $$\frac{75}{\cancel{5}} = \frac{5X}{\cancel{5}}$$ $$15 = X$$ 6. **Final answer:** The rule is $Y = 5X$ and when $Y=75$, $X=15$.