1. **State the problem:** We are given a table of values for $x$ and $y$: $$\begin{array}{c|ccccc} x & 2 & 5 & 6 & 8 & 9 \\ y & 6 & 15 & 18 & 24 & 27 \\\end{array}$$ We want to find the relationship between $x$ and $y$, possibly a function $y=f(x)$ that fits these points. 2. **Check if the relationship is linear:** A linear function has the form $y = mx + b$ where $m$ is the slope and $b$ is the intercept. 3. **Calculate slopes between points:** - Between $(2,6)$ and $(5,15)$: $m = \frac{15-6}{5-2} = \frac{9}{3} = 3$ - Between $(5,15)$ and $(6,18)$: $m = \frac{18-15}{6-5} = \frac{3}{1} = 3$ - Between $(6,18)$ and $(8,24)$: $m = \frac{24-18}{8-6} = \frac{6}{2} = 3$ - Between $(8,24)$ and $(9,27)$: $m = \frac{27-24}{9-8} = \frac{3}{1} = 3$ Since the slope is constant at 3, the relationship is linear. 4. **Find the intercept $b$ using one point, e.g., $(2,6)$:** $$6 = 3 \times 2 + b$$ $$6 = 6 + b$$ $$b = 6 - 6 = 0$$ 5. **Write the function:** $$y = 3x$$ 6. **Verify with other points:** - For $x=5$, $y=3 \times 5 = 15$ (matches table) - For $x=9$, $y=3 \times 9 = 27$ (matches table) **Final answer:** $$y = 3x$$