write and row reduce the augmented matrix to find the general solution:
2x - z = 2
6x + 5y + 3z = 7
2x - y = 4
see all the gory details here:
http://www.gregthatcher.com/Mathematics/GaussJordan.aspx
just enter you matrix elements and it shows all the intermediate reductions.
2 0 -1 2
6 5 3 7
2 -1 0 4
--------------------------
1 0 -1/2 1
0 5 6 1
0 -1 1 2
------------------------------
1 0 -1/2 1
0 1 6/5 1/5
0 0 11/5 11/5
----------------------
1 0 -1/2 1
0 1 6/5 1/5
0 0 1 1
so z = 1
now work back up
get x = 3/2 y = -1 z = 1
To row reduce the augmented matrix and find the general solution, follow these steps:
Step 1: Create the augmented matrix
Write down the coefficients of the unknowns (x, y, and z), as well as the constants on the right-hand side of each equation.
2 -1 -1 | 2
6 5 3 | 7
2 -1 0 | 4
Step 2: Perform row operations to obtain an upper triangular matrix
The goal is to create zeros below the main diagonal.
- Multiply Row 1 by 3 (-3 * Row 1 + Row 2) to eliminate the x term in Row 2.
- Multiply Row 1 by -1 (-1 * Row 1 + Row 3) to eliminate the x term in Row 3.
2 -1 -1 | 2
0 8 6 | 1
0 0 1 | -2
Step 3: Perform back substitution to calculate the values of z, y, and x
Start from the bottom row and work your way up.
From Row 3, we have z = -2.
Substituting z = -2 into Row 2, we get:
8y + 6(-2) = 1
8y = 13
y = 13/8
Finally, substituting y = 13/8 and z = -2 into Row 1, we get:
2x - (-1)(13/8) - (-1)(-2) = 2
2x + 13/8 + 2 = 2
2x + 13/8 = -2
2x = -2 - 13/8
2x = -29/8
x = (-29/8) / 2
x = -29/16
So, the general solution is x = -29/16, y = 13/8, z = -2.