2x1+2x2+2x3=0
-2x1+5x2+2x3=1
8x1+x2+4x3=1
gauss elimnation method
plz sol any one
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x1=5
x2=3
x3=7
To solve the system of equations using Gauss elimination method, follow these steps:
Step 1: Write the augmented matrix for the system of equations:
| 2 1 2 | 0 |
| -2 5 2 | 1 |
| 8 1 4 | 1 |
Step 2: Perform row operations to convert the matrix into row-echelon form or reduced row-echelon form.
Start with the first row:
R1 = R1/2 (Divide the first row by 2)
| 1 0.5 1 | 0 |
| -2 5 2 | 1 |
| 8 1 4 | 1 |
Next, replace R2 with R2 + 2*R1 (Multiply R1 by 2 and add to R2)
| 1 0.5 1 | 0 |
| 0 6 4 | 1 |
| 8 1 4 | 1 |
Similarly, replace R3 with R3 - 8*R1 (Multiply R1 by 8 and subtract from R3)
| 1 0.5 1 | 0 |
| 0 6 4 | 1 |
| 0 -3 -4 | 1 |
Now, divide row 2 by 6:
| 1 0.5 1 | 0 |
| 0 1 2/3 | 1/6 |
| 0 -3 -4 | 1 |
Next, replace row 3 with row 3 + 3*row 2:
| 1 0.5 1 | 0 |
| 0 1 2/3 | 1/6 |
| 0 0 -2/3 | 1/2 |
Step 3: Back-substitution
From the last row, we have -2/3x3 = 1/2, which implies x3 = -3/4.
From the second row, we have x2 + (2/3)(-3/4) = 1/6, which simplifies to x2 = 1/6 + 1/2, x2 = 4/6 or x2 = 2/3.
Finally, from the first row, we have:
x1 + (1/2)(2/3) + (1)(-3/4) = 0,
which simplifies to x1 + 1/3 - 3/4 = 0,
x1 = -1/3 + 3/4,
x1 = 4/12 - 9/12,
x1 = -5/12.
Therefore, the solution to the system of equations is:
x1 = -5/12
x2 = 2/3
x3 = -3/4.
To solve the given system of equations using the Gaussian elimination method, we will eliminate variables to reduce the system into an upper triangular matrix. Here are the steps to solve it:
Step 1: Write the augmented matrix for the system of equations:
[ 2 2 2 | 0 ]
[-2 5 2 | 1 ]
[ 8 1 4 | 1 ]
Step 2: Perform row operations to eliminate the x1 coefficient from the 2nd and 3rd rows.
R2 = R2 + R1 * 1
R3 = R3 - R1 * 4
The updated matrix will be:
[ 2 2 2 | 0 ]
[ 0 7 4 | 1 ]
[ 0 -7 -4 |1 ]
Step 3: Perform row operations to eliminate the x2 coefficient from the 3rd row.
R3 = R3 + R2
The updated matrix will be:
[ 2 2 2 | 0 ]
[ 0 7 4 | 1 ]
[ 0 0 0 | 2 ]
Now, the system is in row-echelon form or upper triangular form.
Step 4: Starting from the bottom row, solve for each variable by back-substitution.
From the last row, 0x1 + 0x2 + 0x3 = 2. This equation is inconsistent (0 = 2), which means there is no solution to the system of equations.
Therefore, the given system of equations has no solution.