1. **State the problem:** We need to find an equation that represents the relationship between $x$ and $y$ given the table: $$\begin{array}{c|cccc} x & 0 & \frac{1}{2} & 1 & \frac{3}{2} \\ y & 0 & 2 & 4 & 6 \\\end{array}$$ 2. **Identify the pattern:** Observe how $y$ changes as $x$ increases. 3. **Check if the relationship is linear:** Calculate the rate of change (slope) between points. Between $x=0$ and $x=\frac{1}{2}$: $$\frac{2 - 0}{\frac{1}{2} - 0} = \frac{2}{\frac{1}{2}} = 4$$ Between $x=\frac{1}{2}$ and $x=1$: $$\frac{4 - 2}{1 - \frac{1}{2}} = \frac{2}{\frac{1}{2}} = 4$$ Between $x=1$ and $x=\frac{3}{2}$: $$\frac{6 - 4}{\frac{3}{2} - 1} = \frac{2}{\frac{1}{2}} = 4$$ The slope is constant at 4, so the relationship is linear. 4. **Use the linear equation formula:** $$y = mx + b$$ where $m$ is the slope and $b$ is the y-intercept. 5. **Find $b$ (y-intercept):** When $x=0$, $y=0$, so $$b = 0$$ 6. **Write the equation:** $$y = 4x + 0$$ or simply $$y = 4x$$ **Final answer:** $$\boxed{y = 4x}$$