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: Rewrite the given simultaneous equations in the standard form ax + by = c. The given equations are: 1) x - 2y + 3 = 0 x - 2y = -3 2) 3y + 2x - 8 = 0 2x + 3y = 8 Step 2: Express the system of equations in matrix form AX = B. 1 & -2 \\ 2 & 3 x \\ y = -3 \\ 8 Here, A = 1 & -2 \\ 2 & 3 , X = x \\ y , and B = -3 \\ 8 . Step 3: Calculate the determinant of matrix A. For a 2 × 2 matrix a & b \\ c & d , the determinant is ad - bc. (A) = (1)(3) - (-2)(2) = 3 - (-4) = 3 + 4 = 7 Step 4: Find the inverse of matrix A, denoted as A^-1. The formula for the inverse of a 2 × 2 matrix is A^-1 = (1)/((A)) d & -b \\ -c & a . A^-1 = (1)/(7) 3 & -(-2) \\ -2 & 1 = (1)/(7) 3 & 2 \\ -2 & 1 Step 5: Solve for X using the formula X = A^-1B. x \\ y = (1)/(7) 3 & 2 \\ -2 & 1 -3 \\ 8 Perform the matrix multiplication: x \\ y = (1)/(7) (3)(-3) + (2)(8) \\ (-2)(-3) + (1)(8) x \\ y = (1)/(7) -9 + 16 \\ 6 + 8 x \\ y = (1)/(7) 7 \\ 14 x \\ y = (7)/(7) \\ (14)/(7) x \\ y = 1 \\ 2 Therefore, x = 1 and y = 2. The solution to the simultaneous equations is: x = 1 y = 2