Geometric Constraints Math Example 1

Solution

1. 1
   Step 1: Circle constraint: $x^2 + y^2 = 25$. Line constraint: $y = x + 1$.
2. 2
   Step 2: Substitute $y = x+1$ into the circle: $x^2 + (x+1)^2 = 25 \Rightarrow x^2 + x^2 + 2x + 1 = 25 \Rightarrow 2x^2 + 2x - 24 = 0 \Rightarrow x^2 + x - 12 = 0$.
3. 3
   Step 3: Factor: $(x+4)(x-3) = 0$, so $x = -4$ or $x = 3$.
4. 4
   Step 4: Corresponding $y$-values: $y = -4+1 = -3$ and $y = 3+1 = 4$. Points: $(-4, -3)$ and $(3, 4)$.