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
To solve the system of equations simultaneously, we first need to simplify the second equation. The given equations are: 1) 6y = x^2 - 2x - 6 2) _n=3^4 x(n-2)^2 = 10y Step 1: Expand the summation in the second equation. The summation runs for n=3 and n=4. For n=3: x(3-2)^2 = x(1)^2 = x For n=4: x(4-2)^2 = x(2)^2 = 4x Adding these terms: x + 4x = 5x. So, the second equation simplifies to: 5x = 10y Step 2: Express y in terms of x from the simplified second equation. Divide both sides by 10: y = (5x)/(10) y = (1)/(2)x Step 3: Substitute the expression for y into the first equation. Substitute y = (1)/(2)x into 6y = x^2 - 2x - 6: 6((1)/(2)x) = x^2 - 2x - 6 3x = x^2 - 2x - 6 Step 4: Rearrange the equation into a standard quadratic form (ax^2 + bx + c = 0). Subtract 3x from both sides: 0 = x^2 - 2x - 3x - 6 0 = x^2 - 5x - 6 Step 5: Solve the quadratic equation for x by factoring. We need two numbers that multiply to -6 and add to -5. These numbers are -6 and 1. (x - 6)(x + 1) = 0 This gives two possible values for x: x - 6 = 0 x = 6 x + 1 = 0 x = -1 Step 6: Find the corresponding y values using y = (1)/(2)x. For x = 6: y = (1)/(2)(6) y = 3 For x = -1: y = (1)/(2)(-1) y = -(1)/(2) The solutions are the pairs (x, y): (6, 3) and (-1, -(1)/(2)) 3 done, 2 left today. You're making progress.