1. Problem: Find which letter corresponds to the solutions of the system \(\{x - y = 3, xy = 18\}\). Formula: To check if a pair \((x,y)\) is a solution, substitute into both equations and verify. Check A (6; 3): - \(6 - 3 = 3\) ✓ - \(6 \times 3 = 18\) ✓ Check B (-3; -6): - \(-3 - (-6) = 3\) ✓ - \(-3 \times -6 = 18\) ✓ Both A and B satisfy the system, so answer is C. 2. Problem: Find \(a\) such that \((a; 3)\) solves \(\{3x + 2y = 12, 5x - 2y = 4\}\). Substitute \(y=3\): - \(3a + 2 \times 3 = 12 \Rightarrow 3a + 6 = 12 \Rightarrow 3a = 6 \Rightarrow a = 2\) Check second equation: - \(5a - 2 \times 3 = 4 \Rightarrow 5 \times 2 - 6 = 4 \Rightarrow 10 - 6 = 4\) ✓ Answer is A. 3. Problem: For system \(\{x^2 + 4x - y = 4, 2x - y = 5\}\), find \(x + y\). From second equation: \(y = 2x - 5\). Substitute into first: - \(x^2 + 4x - (2x - 5) = 4 \Rightarrow x^2 + 4x - 2x + 5 = 4 \Rightarrow x^2 + 2x + 1 = 0\) Factor: - \((x + 1)^2 = 0 \Rightarrow x = -1\) Find \(y\): - \(y = 2(-1) - 5 = -2 - 5 = -7\) Sum: - \(x + y = -1 + (-7) = -8\) Answer is C. 4. Problem: Check if \((1; 2)\) solves \(\{x + 3 = 2y, 4x^2 - 2x = 5y^2 - 63y + 108\}\). Check first: - \(1 + 3 = 4\), \(2 \times 2 = 4\) ✓ Check second: - Left: \(4(1)^2 - 2(1) = 4 - 2 = 2\) - Right: \(5(2)^2 - 63(2) + 108 = 20 - 126 + 108 = 2\) ✓ Both true, answer is C. 5. Problem: Lawn is rectangle with length \(y = 3x\), path 1 m wide around it, path area 36. Path area formula: - \((x + 2)(y + 2) - xy = 36\) because path adds 1 m on each side, total 2 m. Given \(y = 3x\), system: - \(\{(x + 2)(y + 2) - xy = 36, y = 3x\}\) Answer is A. 6. Solve systems: a) \(\{y = x + 3, xy = -2\}\) Substitute \(y\): - \(x(x + 3) = -2 \Rightarrow x^2 + 3x + 2 = 0\) Factor: - \((x + 1)(x + 2) = 0 \Rightarrow x = -1, -2\) Find \(y\): - For \(x = -1\), \(y = 2\) - For \(x = -2\), \(y = 1\) Solutions: \((-1, 2), (-2, 1)\) b) \(\{2x + y = 1, 2x^2 - y = -1.5\}\) From first: \(y = 1 - 2x\) Substitute into second: - \(2x^2 - (1 - 2x) = -1.5 \Rightarrow 2x^2 - 1 + 2x = -1.5\) Simplify: - \(2x^2 + 2x + 0.5 = 0\) Divide by 0.5: - \(4x^2 + 4x + 1 = 0\) Discriminant: - \(16 - 16 = 0\), one root: - \(x = -\frac{4}{8} = -0.5\) Find \(y\): - \(y = 1 - 2(-0.5) = 1 + 1 = 2\) Solution: \((-0.5, 2)\) c) \(\{x^2 + y^2 = 9, y = x + 3\}\) Substitute: - \(x^2 + (x + 3)^2 = 9 \Rightarrow x^2 + x^2 + 6x + 9 = 9\) Simplify: - \(2x^2 + 6x = 0 \Rightarrow 2x(x + 3) = 0\) Roots: - \(x = 0, -3\) Find \(y\): - \(x=0 \Rightarrow y=3\) - \(x=-3 \Rightarrow y=0\) Solutions: \((0,3), (-3,0)\) d) \(\{(x - 2)^2 + (y - 1)^2 = 10, 2(x - y) = 3(x - y) + 3\}\) Rewrite second: - \(2(x - y) - 3(x - y) = 3 \Rightarrow - (x - y) = 3 \Rightarrow y - x = 3\) So \(y = x + 3\) Substitute into first: - \((x - 2)^2 + (x + 3 - 1)^2 = 10 \Rightarrow (x - 2)^2 + (x + 2)^2 = 10\) Expand: - \(x^2 - 4x + 4 + x^2 + 4x + 4 = 10\) Simplify: - \(2x^2 + 8 = 10 \Rightarrow 2x^2 = 2 \Rightarrow x^2 = 1\) Roots: - \(x = \pm 1\) Find \(y\): - \(x=1 \Rightarrow y=4\) - \(x=-1 \Rightarrow y=2\) Solutions: \((1,4), (-1,2)\)