1. **State the problem:** We need to find the values of $a$ and $b$ such that the midpoint of the points $(a, 3)$ and $(7, b)$ is $(5, 4)$. 2. **Formula for midpoint:** The midpoint $M$ of two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$$ 3. **Apply the formula:** Given midpoint is $(5, 4)$, so: $$\frac{a + 7}{2} = 5 \quad \text{and} \quad \frac{3 + b}{2} = 4$$ 4. **Solve for $a$:** $$\frac{a + 7}{2} = 5$$ Multiply both sides by 2: $$\cancel{2} \times \frac{a + 7}{\cancel{2}} = 5 \times 2$$ $$a + 7 = 10$$ Subtract 7 from both sides: $$a = 10 - 7 = 3$$ 5. **Solve for $b$:** $$\frac{3 + b}{2} = 4$$ Multiply both sides by 2: $$\cancel{2} \times \frac{3 + b}{\cancel{2}} = 4 \times 2$$ $$3 + b = 8$$ Subtract 3 from both sides: $$b = 8 - 3 = 5$$ **Final answer:** $$a = 3, \quad b = 5$$