This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Write down the given simultaneous equations. 2x + y = 5 (*1) x^2 + y^2 = 25 (*2) Step 2: Express y in terms of x from the linear equation (*1). From equation (*1): y = 5 - 2x Step 3: Substitute this expression for y into the quadratic equation (*2). x^2 + (5 - 2x)^2 = 25 Step 4: Expand and simplify the equation. x^2 + (5^2 - 2(5)(2x) + (2x)^2) = 25 x^2 + (25 - 20x + 4x^2) = 25 Combine like terms: 5x^2 - 20x + 25 = 25 Subtract 25 from both sides: 5x^2 - 20x = 0 Step 5: Solve the quadratic equation for x. Factor out 5x: 5x(x - 4) = 0 This gives two possible solutions for x: 5x = 0 x = 0 x - 4 = 0 x = 4 Step 6: Substitute each value of x back into the expression for y (y = 5 - 2x) to find the corresponding y values. For x = 0: y = 5 - 2(0) y = 5 - 0 y = 5 One solution is (x, y) = (0, 5). For x = 4: y = 5 - 2(4) y = 5 - 8 y = -3 The second solution is (x, y) = (4, -3). The solutions to the simultaneous equations are x=0, y=5 and x=4, y=-3. 3 done, 2 left today. You're making progress.