Solve each equation using augmented matrices.
2x+y-2z=7
x-2y-5z=-1
4x+y+z=-1
To solve the given system of equations using augmented matrices, we will create the augmented matrix and then perform row operations to achieve row-echelon form or reduced row-echelon form.
Step 1: Create the augmented matrix
The augmented matrix is created by writing down the coefficients of the variables and the constants in a matrix form. For the given system of equations:
2x + y - 2z = 7
x - 2y - 5z = -1
4x + y + z = -1
The augmented matrix [A|b] would be:
| 2 1 -2 | 7 |
| 1 -2 -5 | -1 |
| 4 1 1 | -1 |
Step 2: Perform row operations
The goal is to transform the augmented matrix into row-echelon form or reduced row-echelon form. Here, we will use Gaussian elimination to achieve this.
To begin, we can interchange rows if necessary to bring a non-zero entry to the upper-left corner of the matrix. In this case, we can swap the first and second rows since the first entry of the second row is non-zero.
| 1 -2 -5 | -1 |
| 2 1 -2 | 7 |
| 4 1 1 | -1 |
Next, we use row operations to create zeros below the leading entry (pivot) in the first column.
Multiply the first row by 2 and subtract it from the second row:
| 1 -2 -5 | -1 |
| 0 5 8 | 9 |
| 4 1 1 | -1 |
Multiply the first row by 4 and subtract it from the third row:
| 1 -2 -5 | -1 |
| 0 5 8 | 9 |
| 0 9 21 | 3 |
Next, we need to create zeros below the leading entry in the second column.
Multiply the second row by 9 and subtract it from the third row:
| 1 -2 -5 | -1 |
| 0 5 8 | 9 |
| 0 0 -3 | -75 |
Finally, we need to make the leading entry in each row equal to 1 by using row operations.
Divide the second row by 5:
| 1 -2 -5 | -1 |
| 0 1 8/5 | 9/5 |
| 0 0 -3 | -75 |
Multiply the third row by -1/3:
| 1 -2 -5 | -1 |
| 0 1 8/5 | 9/5 |
| 0 0 1 | 25 |
Step 3: Back substitution
In the row-echelon form, we can start from the bottom row and substitute the values back into the equations above to solve for the variables.
From the third equation, we have z = 25.
Substitute the value of z into the second equation, we have:
y + (8/5)(25) = 9/5
y + 40 = 9/5
y = 9/5 - 40 = -191/5
Substitute the values of z and y into the first equation, we have:
x - 2(-191/5) - 5(25) = -1
x + 382/5 - 125 = -1
x = -1 - 382/5 + 125 = -57/5
So, the solution to the system of equations is:
x = -57/5,
y = -191/5,
z = 25.