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 Cramer's Rule: 1) x + 2y + 3z = 1 2) 3x + 2y + z = 4 3) x + 3y + 2z = 0 Step 1: Write the coefficient matrix A and the constant vector B. A = 1 & 2 & 3 \\ 3 & 2 & 1 \\ 1 & 3 & 2 and B = 1 \\ 4 \\ 0 Step 2: Calculate the determinant of the coefficient matrix, D. D = (A) = 1 & 2 & 3 \\ 3 & 2 & 1 \\ 1 & 3 & 2 D = 1 2 & 1 \\ 3 & 2 - 2 3 & 1 \\ 1 & 2 + 3 3 & 2 \\ 1 & 3 D = 1((2)(2) - (1)(3)) - 2((3)(2) - (1)(1)) + 3((3)(3) - (2)(1)) D = 1(4 - 3) - 2(6 - 1) + 3(9 - 2) D = 1(1) - 2(5) + 3(7) D = 1 - 10 + 21 = 12 Step 3: Calculate D_x by replacing the first column of A with B. D_x = 1 & 2 & 3 \\ 4 & 2 & 1 \\ 0 & 3 & 2 D_x = 1 2 & 1 \\ 3 & 2 - 2 4 & 1 \\ 0 & 2 + 3 4 & 2 \\ 0 & 3 D_x = 1((2)(2) - (1)(3)) - 2((4)(2) - (1)(0)) + 3((4)(3) - (2)(0)) D_x = 1(4 - 3) - 2(8 - 0) + 3(12 - 0) D_x = 1(1) - 2(8) + 3(12) D_x = 1 - 16 + 36 = 21 Step 4: Calculate D_y by replacing the second column of A with B. D_y = 1 & 1 & 3 \\ 3 & 4 & 1 \\ 1 & 0 & 2 D_y = 1 4 & 1 \\ 0 & 2 - 1 3 & 1 \\ 1 & 2 + 3 3 & 4 \\ 1 & 0 D_y = 1((4)(2) - (1)(0)) - 1((3)(2) - (1)(1)) + 3((3)(0) - (4)(1)) D_y = 1(8 - 0) - 1(6 - 1) + 3(0 - 4) D_y = 1(8) - 1(5) + 3(-4) D_y = 8 - 5 - 12 = -9 Step 5: Calculate D_z by replacing the third column of A with B. D_z = 1 & 2 & 1 \\ 3 & 2 & 4 \\ 1 & 3 & 0 D_z = 1 2 & 4 \\ 3 & 0 - 2 3 & 4 \\ 1 & 0 + 1 3 & 2 \\ 1 & 3 D_z = 1((2)(0) - (4)(3)) - 2((3)(0) - (4)(1)) + 1((3)(3) - (2)(1)) D_z = 1(0 - 12) - 2(0 - 4) + 1(9 - 2) D_z = 1(-12) - 2(-4) + 1(7) D_z = -12 + 8 + 7 = 3 Step 6: Apply Cramer's Rule to find x, y, z. x = (D_x)/(D) = (21)/(12) = (7)/(4) y = (D_y)/(D) = (-9)/(12) = -(3)/(4) z = (D_z)/(D) = (3)/(12) = (1)/(4) The solution is: x = (7)/(4) y = -(3)/(4) z = (1)/(4) 3 done, 2 left today. You're making progress.