1. The problem states there is a proportional relationship between the number of books $x$ and the shelf space $y$ in inches. 2. The constant of proportionality $k$ in a direct proportionality is given by the formula: $$y = kx$$ 3. From the graph, when $x = 10$, $y = 10$. 4. Substitute these values into the formula to find $k$: $$10 = k \times 10$$ 5. Divide both sides by 10 to isolate $k$: $$\frac{10}{\cancel{10}} = k \times \frac{\cancel{10}}{10}$$ $$1 = k$$ 6. Therefore, the constant of proportionality is $1$. This means each book takes up exactly 1 inch of shelf space.