Solve the system of equations using matrices.Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. begin{cases}x+y-z=-22x-y+z=5-x+2y+2z=1end{cases}

Question
Matrices
asked 2021-01-08
Solve the system of equations using matrices.Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
\(\begin{cases}x+y-z=-2\\2x-y+z=5\\-x+2y+2z=1\end{cases}\)

Answers (1)

2021-01-09
Step 1
Given:
The system of linear equations are,
\(x+y-z=-2\)
\(2x-y+z=5\)
\(-x+2y+2z=1\)
For applying Gauss-Jordan elimination method, the above system of equations can be represented in matrix form as,
\(\begin{bmatrix}1 & 1&-1&-2 \\2 & -1&1&5\\-1&2&2&1 \end{bmatrix}\)
Step 2
The above matrix can be converted into row echelon form as,
\(\begin{bmatrix}1 & 1&-1&-2 \\0 & -3&3&9\\-1&2&2&1 \end{bmatrix} \left(\text{By applying: } R_2 \rightarrow R_2-2R_1\right)\)
\(\begin{bmatrix}1 & 1&-1&-2 \\0 & -3&3&9\\0&3&1&-1 \end{bmatrix}\left(\text{By applying: } R_3 \rightarrow R_3+R_1\right)\)
\(\begin{bmatrix}1 & 1&-1&-2 \\0 & 1&-1&-3\\0&3&1&-1 \end{bmatrix} \left(\text{By applying: } R_2 \rightarrow \frac{-1}{3}R_2\right)\)
\(\begin{bmatrix}1 & 1&-1&-2 \\0 & 1&-1&-3\\0&0&4&8 \end{bmatrix} \left(\text{By applying: } R_3 \rightarrow R_3-3R_2\right)\)
Step 3
\(\begin{bmatrix}1 & 1&-1&-2 \\0 & 1&-1&-3\\0&0&1&2 \end{bmatrix} \ \ \left(\text{By applying: } R_3 \rightarrow \frac{1}{4}R_3\right)\)
\(\begin{bmatrix}1 & 1&-1&-2 \\0 & 1&0&-1\\0&0&1&2 \end{bmatrix} \ \ \left(\text{By applying: } R_2 \rightarrow R_2+R_3\right)\)
\(\begin{bmatrix}1 & 1&0&0 \\0 & 1&0&-1\\0&0&1&2 \end{bmatrix} \ \ \left(\text{By applying: } R_1 \rightarrow R_1+R_3\right)\)
Step 4
\(\begin{bmatrix}1 & 0&0&1 \\0 & 1&0&-1\\0&0&1&2 \end{bmatrix} \ \ \left(\text{By applying: } R_1 \rightarrow R_1-R_2\right)\)
Hence, the solutions of the given system of equations are x=1 , y=-1 and z=2
0

Relevant Questions

asked 2021-02-08
Solve the system of given equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
\(\begin{cases}x+3y=0\\x+y+z=1\\3x-y-z=11\end{cases}\)
asked 2021-01-31
Solve the system of given equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
\(\begin{cases}3a-b-4c=3\\2a-b+2c=-8\\a+2b-3c=9\end{cases}\)
asked 2020-11-23
Solve the system of equations using matrices. Use the Gaussian elimination method with​ back-substitution.
x+4y=0 x+5y+z=1 5x-y-z=79
asked 2020-12-30
Solve the shown using matrices:
\(\begin{cases}2x-y-2z=-1\\ x-2y-z=1\\ x+y+z=1 \end{cases}\)
asked 2021-02-05
Solve the systems of equations using matrices.
4x+5y=8
3x-4y=3
Leave answer in fraction form.
4x+y+z=3
-x+y=-11+2z
2y+2z=-1-x
asked 2021-03-06
Solve the systems of equations using matrices.
4x+y+z=3
-x+y-2z=-11
x+2y+2z=-1
asked 2021-02-13
Solve the system of linear equations using matrices.
x+y+z=3
2x+3y+2z=7
3x-4y+z=4
asked 2021-02-19
Use the​ Gauss-Jordan method to solve the system of equations. If the system has infinitely many​ solutions, give the solution with z arbitrary.
x-5y+2z=1
3x-4y+2z=-1
asked 2021-01-19
Use Gaussian elimination to find the complete solution to the system of given equations, or show that none exists.
\(\displaystyle{\left\lbrace\begin{array}{c} {2}{x}-{4}{y}+{z}={3}\\{x}-{3}{y}+{z}={5}\\{3}{x}-{7}{y}+{2}{z}={12}\end{array}\right.}\)
asked 2021-02-02
Solve the system of equations (Use matrices.):
x-2y+z = 16,
2x-y-z = 14,
3x+5y-4z =-10
...