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
Here's how to solve the system of equations: The given equations are: 1) x + 3y = 13 2) x^2 + 3y^2 = 43 Step 1: Express x in terms of y from equation (1). x = 13 - 3y Step 2: Substitute this expression for x into equation (2). (13 - 3y)^2 + 3y^2 = 43 Step 3: Expand and simplify the equation. (13^2 - 2 · 13 · 3y + (3y)^2) + 3y^2 = 43 169 - 78y + 9y^2 + 3y^2 = 43 12y^2 - 78y + 169 - 43 = 0 12y^2 - 78y + 126 = 0 Step 4: Divide the entire equation by 6 to simplify. (12y^2)/(6) - (78y)/(6) + (126)/(6) = 0 2y^2 - 13y + 21 = 0 Step 5: Solve the quadratic equation for y. We can factor this quadratic equation. We need two numbers that multiply to 2 × 21 = 42 and add up to -13. These numbers are -6 and -7. 2y^2 - 6y - 7y + 21 = 0 2y(y - 3) - 7(y - 3) = 0 (2y - 7)(y - 3) = 0 This gives two possible values for y: 2y - 7 = 0 2y = 7 y = (7)/(2) y - 3 = 0 y = 3 Step 6: Substitute these values of y back into the expression for x (x = 13 - 3y) to find the corresponding x values. Case 1: When y = 3 x = 13 - 3(3) x = 13 - 9 x = 4 Case 2: When y = (7)/(2) x = 13 - 3((7)/(2)) x = 13 - (21)/(2) x = (26)/(2) - (21)/(2) x = (5)/(2) The solutions are the pairs (x, y): (4, 3) and ((5)/(2), (7)/(2))