1. **Stating the problem:** Find the coordinates of the points of intersection of the line $x + y = 4$ and the circle $x^2 + y^2 = 10$. 2. **Formula and approach:** To find the intersection points, substitute the expression for $x$ or $y$ from the line equation into the circle equation and solve for the remaining variable. 3. From the line equation, express $x$ in terms of $y$: $$x = 4 - y$$ 4. Substitute $x = 4 - y$ into the circle equation: $$ (4 - y)^2 + y^2 = 10 $$ 5. Expand and simplify: $$ (4 - y)^2 + y^2 = 10 $$ $$ 16 - 8y + y^2 + y^2 = 10 $$ $$ 16 - 8y + 2y^2 = 10 $$ 6. Rearrange to standard quadratic form: $$ 2y^2 - 8y + 16 - 10 = 0 $$ $$ 2y^2 - 8y + 6 = 0 $$ 7. Simplify by dividing all terms by 2: $$ \cancel{2}y^2 - \cancel{8}y + \cancel{6} = 0 $$ $$ y^2 - 4y + 3 = 0 $$ 8. Factor the quadratic: $$ (y - 3)(y - 1) = 0 $$ 9. Solve for $y$: $$ y = 3 \quad \text{or} \quad y = 1 $$ 10. Find corresponding $x$ values using $x = 4 - y$: - For $y = 3$: $x = 4 - 3 = 1$ - For $y = 1$: $x = 4 - 1 = 3$ 11. **Final answer:** The points of intersection are: $$ (1, 3) \quad \text{and} \quad (3, 1) $$