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: Calculate the determinant of the left-hand side matrix. The determinant of a 2 × 2 matrix a & b \\ c & d is ad - bc. For the left-hand side: -2n+3 & n+1 \\ 2n & n = (-2n+3)(n) - (n+1)(2n) Expand the terms: (-2n^2 + 3n) - (2n^2 + 2n) -2n^2 + 3n - 2n^2 - 2n -4n^2 + n Step 2: Calculate the determinant of the right-hand side matrix. For the right-hand side: -2n+5 & 4n \\ -n+2 & 3n = (-2n+5)(3n) - (4n)(-n+2) Expand the terms: (-6n^2 + 15n) - (-4n^2 + 8n) -6n^2 + 15n + 4n^2 - 8n -2n^2 + 7n Step 3: Set the two determinants equal to each other and solve for n. -4n^2 + n = -2n^2 + 7n Move all terms to one side to form a quadratic equation: 0 = -2n^2 + 7n + 4n^2 - n 0 = 2n^2 + 6n Factor out 2n: 2n(n + 3) = 0 This gives two possible solutions for n: 2n = 0 n = 0 n + 3 = 0 n = -3 The solutions for n are 0 and -3. n = 0, -3 That's 2 down. 3 left today — send the next one.