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
Okay, let's solve the system of equations using the augmented matrix form and Gaussian elimination (Gauss-Jordan method). The given system of equations is: 1) 2x + 2y - z = -7 2) x - y + 2z = 4 3) 3x + 3y - z = 0 Step 1: Write the system as an augmented matrix. 2 & 2 & -1 & | & -7 \\ 1 & -1 & 2 & | & 4 \\ 3 & 3 & -1 & | & 0 Step 2: Perform row operations to transform the matrix into reduced row-echelon form. Swap R_1 and R_2 to get a leading 1 in the first row: R_1 R_2 \\ 1 & -1 & 2 & | & 4 \\ 2 & 2 & -1 & | & -7 \\ 3 & 3 & -1 & | & 0 Make the elements below the leading 1 in the first column zero: R_2 - 2R_1 R_2 \\ R_3 - 3R_1 R_3 \\ 1 & -1 & 2 & | & 4 \\ 0 & 4 & -5 & | & -15 \\ 0 & 6 & -7 & | & -12 Get a leading 1 in the second row, second column: (1)/(4)R_2 R_2 \\ 1 & -1 & 2 & | & 4 \\ 0 & 1 & -(5)/(4) & | & -(15)/(4) \\ 0 & 6 & -7 & | & -12 Make the element below the leading 1 in the second column zero: R_3 - 6R_2 R_3 \\ 1 & -1 & 2 & | & 4 \\ 0 & 1 & -(5)/(4) & | & -(15)/(4) \\ 0 & 0 & (1)/(2) & | & (21)/(2) Get a leading 1 in the third row, third column: 2R_3 R_3 \\ 1 & -1 & 2 & | & 4 \\ 0 & 1 & -(5)/(4) & | & -(15)/(4) \\ 0 & 0 & 1 & | & 21 Make the elements above the leading 1 in the third column zero: R_2 + (5)/(4)R_3 R_2 \\ R_1 - 2R_3 R_1 \\ 1 & -1 & 0 & | & -38 \\ 0 & 1 & 0 & | & (45)/(2) \\ 0 & 0 & 1 & | & 21 Make the element above the leading 1 in the second column zero: R_1 + R_2 R_1 \\ 1 & 0 & 0 & | & -(31)/(2) \\ 0 & 1 & 0 & | & (45)/(2) \\ 0 & 0 & 1 & | & 21 Step 3: Convert the reduced row-echelon form back into a system of equations. From the matrix, we can directly read the solution: x = -(31)/(2) y = (45)/(2) z = 21 x = -(31)/(2), y = (45)/(2), z = 21 That's 2 down. 3 left today — send the next one.