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 using the matrix method, we first write the system in the form AX = B. The given equations are: 1) 3x + y = 9 2) 2x - y = 1 Step 1: Write the system in matrix form. The coefficient matrix A, the variable matrix X, and the constant matrix B are: A = 3 & 1 \\ 2 & -1 , X = x \\ y , B = 9 \\ 1 So, the matrix equation is: 3 & 1 \\ 2 & -1 x \\ y = 9 \\ 1 Step 2: Find the determinant of matrix A. For a 2 × 2 matrix a & b \\ c & d , the determinant is ad - bc. (A) = (3)(-1) - (1)(2) = -3 - 2 = -5 Step 3: Find the inverse of matrix A. For a 2 × 2 matrix a & b \\ c & d , the inverse is A^-1 = (1)/((A)) d & -b \\ -c & a . A^-1 = (1)/(-5) -1 & -1 \\ -2 & 3 A^-1 = (-1)/(-5) & (-1)/(-5) \\ (-2)/(-5) & (3)/(-5) = (1)/(5) & (1)/(5) \\ (2)/(5) & -(3)/(5) Step 4: Solve for X using X = A^-1B. x \\ y = (1)/(5) & (1)/(5) \\ (2)/(5) & -(3)/(5) 9 \\ 1 Perform the matrix multiplication: x = ((1)/(5))(9) + ((1)/(5))(1) = (9)/(5) + (1)/(5) = (10)/(5) = 2 y = ((2)/(5))(9) + (-(3)/(5))(1) = (18)/(5) - (3)/(5) = (15)/(5) = 3 Thus, the solution is x=2 and y=3. The final answer is x=2, y=3.