1. **Problem statement:** Find the ratio in which the line $2x + y - 4 = 0$ divides the line segment joining points $A(2, -2)$ and $B(3, 7)$. 2. **Formula used:** If a point $P$ divides the segment $AB$ in the ratio $k:1$, then the coordinates of $P$ are given by the section formula: $$P = \left( \frac{kx_2 + x_1}{k+1}, \frac{ky_2 + y_1}{k+1} \right)$$ where $A = (x_1, y_1)$ and $B = (x_2, y_2)$. 3. **Step:** Let the point of division be $P(x, y)$ lying on the line $2x + y - 4 = 0$. Using the section formula: $$x = \frac{3k + 2}{k+1}, \quad y = \frac{7k - 2}{k+1}$$ 4. **Step:** Substitute $x$ and $y$ into the line equation: $$2x + y - 4 = 0$$ $$2 \times \frac{3k + 2}{k+1} + \frac{7k - 2}{k+1} - 4 = 0$$ 5. **Step:** Multiply through by $k+1$ to clear the denominator: $$2(3k + 2) + (7k - 2) - 4(k+1) = 0$$ $$6k + 4 + 7k - 2 - 4k - 4 = 0$$ 6. **Step:** Simplify: $$6k + 7k - 4k + 4 - 2 - 4 = 0$$ $$9k - 2 = 0$$ 7. **Step:** Solve for $k$: $$9k = 2$$ $$k = \frac{2}{9}$$ **Answer:** The line divides the segment $AB$ in the ratio $\boxed{2:9}$.