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
5. Find the inverse of the matrix 1 & 1 \\ 3 & 1 . Hence, determine the point of intersection of the lines y+x=7 and 3x+y=15. Step 1: Find the inverse of the matrix A = 1 & 1 \\ 3 & 1 . The determinant of A is given by (A) = (1)(1) - (1)(3) = 1 - 3 = -2. The inverse matrix A^-1 is calculated as: A^-1 = (1)/((A)) d & -b \\ -c & a A^-1 = (1)/(-2) 1 & -1 \\ -3 & 1 A^-1 = -(1)/(2) & (1)/(2) \\ (3)/(2) & -(1)/(2) The inverse of the matrix is -(1)/(2) & (1)/(2) \\ (3)/(2) & -(1)/(2) . Step 2: Write the system of linear equations in matrix form. The given equations are: x+y=7 3x+y=15 This can be written as AX=B: 1 & 1 \\ 3 & 1 x \\ y = 7 \\ 15 Step 3: Use the inverse matrix to solve for X. X = A^-1B x \\ y = -(1)/(2) & (1)/(2) \\ (3)/(2) & -(1)/(2) 7 \\ 15 Step 4: Perform the matrix multiplication. x = (-(1)/(2))(7) + ((1)/(2))(15) = -(7)/(2) + (15)/(2) = (8)/(2) = 4 y = ((3)/(2))(7) + (-(1)/(2))(15) = (21)/(2) - (15)/(2) = (6)/(2) = 3 The point of intersection is (x, y) = (4, 3). That's 2 down. 3 left today — send the next one.