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To solve the given system of linear equations using Gaussian elimination with back-substitution, we will follow these steps:
1. Write the augmented matrix for the system of equations:
[tex]\[ \begin{aligned} 3x - 2y + z &= 11 \\ -x + y + 2z &= -11 \\ x - y - 4z &= 17 \end{aligned} \][/tex]
The augmented matrix representing this system is:
[tex]\[ \left[\begin{array}{ccc|c} 3 & -2 & 1 & 11 \\ -1 & 1 & 2 & -11 \\ 1 & -1 & -4 & 17 \end{array}\right] \][/tex]
2. Perform Gaussian elimination to transform the matrix into an upper triangular form.
- Normalize the first row by dividing by the pivot element (3):
[tex]\[ \left[\begin{array}{ccc|c} 1 & -0.6667 & 0.3333 & 3.6667 \\ -1 & 1 & 2 & -11 \\ 1 & -1 & -4 & 17 \end{array}\right] \][/tex]
- Eliminate the entries below the pivot in column 1:
[tex]\[ \left[\begin{array}{ccc|c} 1 & -0.6667 & 0.3333 & 3.6667 \\ 0 & 0.3333 & 2.3333 & -7.3333 \\ 0 & -0.3333 & -4.3333 & 13.3333 \end{array}\right] \][/tex]
- Normalize the second row by dividing by the pivot element (0.3333):
[tex]\[ \left[\begin{array}{ccc|c} 1 & -0.6667 & 0.3333 & 3.6667 \\ 0 & 1 & 7 & -22 \\ 0 & -0.3333 & -4.3333 & 13.3333 \end{array}\right] \][/tex]
- Eliminate the entries below the pivot in column 2:
[tex]\[ \left[\begin{array}{ccc|c} 1 & -0.6667 & 0.3333 & 3.6667 \\ 0 & 1 & 7 & -22 \\ 0 & 0 & -2 & 6 \end{array}\right] \][/tex]
- Normalize the third row by dividing by the pivot element (-2):
[tex]\[ \left[\begin{array}{ccc|c} 1 & -0.6667 & 0.3333 & 3.6667 \\ 0 & 1 & 7 & -22 \\ 0 & 0 & 1 & -3 \end{array}\right] \][/tex]
3. Perform back-substitution to find the values of [tex]\(x, y,\)[/tex] and [tex]\(z\)[/tex].
- From the third row:
[tex]\[ z = -3 \][/tex]
- Substitute [tex]\(z = -3\)[/tex] into the second row equation:
[tex]\[ y + 7(-3) = -22 \implies y - 21 = -22 \implies y = -1 \][/tex]
- Substitute [tex]\( y = -1 \)[/tex] and [tex]\( z = -3 \)[/tex] into the first row equation:
[tex]\[ x - 0.6667(-1) + 0.3333(-3) = 3.6667 \implies x + 0.6667 - 1 = 3.6667 \implies x + 0.6667 = 4.6667 \implies x = 4 \][/tex]
So, the solution to the system of equations is:
[tex]\[ (x, y, z) = (4, -1, -3) \][/tex]
1. Write the augmented matrix for the system of equations:
[tex]\[ \begin{aligned} 3x - 2y + z &= 11 \\ -x + y + 2z &= -11 \\ x - y - 4z &= 17 \end{aligned} \][/tex]
The augmented matrix representing this system is:
[tex]\[ \left[\begin{array}{ccc|c} 3 & -2 & 1 & 11 \\ -1 & 1 & 2 & -11 \\ 1 & -1 & -4 & 17 \end{array}\right] \][/tex]
2. Perform Gaussian elimination to transform the matrix into an upper triangular form.
- Normalize the first row by dividing by the pivot element (3):
[tex]\[ \left[\begin{array}{ccc|c} 1 & -0.6667 & 0.3333 & 3.6667 \\ -1 & 1 & 2 & -11 \\ 1 & -1 & -4 & 17 \end{array}\right] \][/tex]
- Eliminate the entries below the pivot in column 1:
[tex]\[ \left[\begin{array}{ccc|c} 1 & -0.6667 & 0.3333 & 3.6667 \\ 0 & 0.3333 & 2.3333 & -7.3333 \\ 0 & -0.3333 & -4.3333 & 13.3333 \end{array}\right] \][/tex]
- Normalize the second row by dividing by the pivot element (0.3333):
[tex]\[ \left[\begin{array}{ccc|c} 1 & -0.6667 & 0.3333 & 3.6667 \\ 0 & 1 & 7 & -22 \\ 0 & -0.3333 & -4.3333 & 13.3333 \end{array}\right] \][/tex]
- Eliminate the entries below the pivot in column 2:
[tex]\[ \left[\begin{array}{ccc|c} 1 & -0.6667 & 0.3333 & 3.6667 \\ 0 & 1 & 7 & -22 \\ 0 & 0 & -2 & 6 \end{array}\right] \][/tex]
- Normalize the third row by dividing by the pivot element (-2):
[tex]\[ \left[\begin{array}{ccc|c} 1 & -0.6667 & 0.3333 & 3.6667 \\ 0 & 1 & 7 & -22 \\ 0 & 0 & 1 & -3 \end{array}\right] \][/tex]
3. Perform back-substitution to find the values of [tex]\(x, y,\)[/tex] and [tex]\(z\)[/tex].
- From the third row:
[tex]\[ z = -3 \][/tex]
- Substitute [tex]\(z = -3\)[/tex] into the second row equation:
[tex]\[ y + 7(-3) = -22 \implies y - 21 = -22 \implies y = -1 \][/tex]
- Substitute [tex]\( y = -1 \)[/tex] and [tex]\( z = -3 \)[/tex] into the first row equation:
[tex]\[ x - 0.6667(-1) + 0.3333(-3) = 3.6667 \implies x + 0.6667 - 1 = 3.6667 \implies x + 0.6667 = 4.6667 \implies x = 4 \][/tex]
So, the solution to the system of equations is:
[tex]\[ (x, y, z) = (4, -1, -3) \][/tex]
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