1. **State the problem:** We need to find the ratio of the change in $x$ to the change in $y$ between the points $(2,1)$ and $(4,7)$ on the line. 2. **Recall the formula for change:** The change in $x$ is $\Delta x = x_2 - x_1$ and the change in $y$ is $\Delta y = y_2 - y_1$. 3. **Calculate the changes:** $$\Delta x = 4 - 2 = 2$$ $$\Delta y = 7 - 1 = 6$$ 4. **Form the ratio of change in $x$ to change in $y$:** $$\text{Ratio} = 1 : \frac{\Delta y}{\Delta x} = 1 : \frac{6}{2}$$ 5. **Simplify the ratio:** $$1 : \frac{\cancel{6}}{\cancel{2}} = 1 : 3$$ 6. **Interpretation:** This means for every 1 unit change in $x$, there is a 3 unit change in $y$. **Final answer:** The ratio of the change in $x$ to the change in $y$ is $1 : 3$.