Problem
For each table, determine whether it shows a direct variation, an inverse variation, or neither. Write the equation for the direct or inverse variation when it exists. (a) \begin{tabular}{|c|c|} \hline$x$ & $y$ \\ \hline 2 & 5 \\ \hline 6 & 15 \\ \hline 8 & 20 \\ \hline \end{tabular} (b) \begin{tabular}{|c|c|} \hline$x$ & $y$ \\ \hline 2 & 7 \\ \hline 4 & 3.5 \\ \hline 8 & 1.75 \\ \hline \end{tabular} Direct variation Equation: Equation: Inverse variation Equation: Neither Direct variation Equation: $\square$ Equation: $\square$ Neither
Solution
Step 1 ：This table shows a direct variation. As x increases, y also increases. Moreover, the ratio of y to x is constant (\(\frac{5}{2} = \frac{15}{6} = \frac{20}{8} = 2.5\)). Therefore, the equation for the direct variation is \(y = 2.5x\).
Step 2 ：This table shows an inverse variation. As x increases, y decreases. Moreover, the product of x and y is constant (\(2*7 = 4*3.5 = 8*1.75 = 14\)). Therefore, the equation for the inverse variation is \(y = \frac{14}{x}\).
Step 3 ：The equation for the direct variation is \(\boxed{y = 2.5x}\).
Step 4 ：The equation for the inverse variation is \(\boxed{y = \frac{14}{x}}\).
