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, we first write the equations in standard form ax + by + cz = d: 1) 2x + 0y + 3z = -1 2) 0x + 1y + 2z = 5 3) 1x + 1y + 0z = 1 Step 1: Form the coefficient matrix A and calculate its determinant, (A). A = 2 & 0 & 3 \\ 0 & 1 & 2 \\ 1 & 1 & 0 (A) = 2(1 · 0 - 2 · 1) - 0(0 · 0 - 2 · 1) + 3(0 · 1 - 1 · 1) (A) = 2(-2) - 0 + 3(-1) (A) = -4 - 3 = -7 Step 2: Form the matrix A_x by replacing the first column of A with the constant terms and calculate its determinant, (A_x). A_x = -1 & 0 & 3 \\ 5 & 1 & 2 \\ 1 & 1 & 0 (A_x) = -1(1 · 0 - 2 · 1) - 0(5 · 0 - 2 · 1) + 3(5 · 1 - 1 · 1) (A_x) = -1(-2) - 0 + 3(4) (A_x) = 2 + 12 = 14 Step 3: Form the matrix A_y by replacing the second column of A with the constant terms and calculate its determinant, (A_y). A_y = 2 & -1 & 3 \\ 0 & 5 & 2 \\ 1 & 1 & 0 (A_y) = 2(5 · 0 - 2 · 1) - (-1)(0 · 0 - 2 · 1) + 3(0 · 1 - 5 · 1) (A_y) = 2(-2) + 1(-2) + 3(-5) (A_y) = -4 - 2 - 15 = -21 Step 4: Form the matrix A_z by replacing the third column of A with the constant terms and calculate its determinant, (A_z). A_z = 2 & 0 & -1 \\ 0 & 1 & 5 \\ 1 & 1 & 1 (A_z) = 2(1 · 1 - 5 · 1) - 0(0 · 1 - 5 · 1) + (-1)(0 · 1 - 1 · 1) (A_z) = 2(-4) - 0 - 1(-1) (A_z) = -8 + 1 = -7 Step 5: Calculate the values of x, y, and z using Cramer's rule. x = ((A_x))/((A)) = (14)/(-7) = -2 y = ((A_y))/((A)) = (-21)/(-7) = 3 z = ((A_z))/((A)) = (-7)/(-7) = 1 The values are: x = -2, y = 3, z = 1 Send me the next one 📸