which of the following elementary row operations could be used to cancel the z term from equation 2 in the following system of equations
X+y+z=1 (eq1)
X+y-2z=-2 (eq2)
3x-2y+z=6 (eq3)
how about : double the first, then add it to the second
To cancel the z term from equation 2 in the given system of equations, we need to perform elementary row operations on the augmented matrix representing the system.
First, let's write the augmented matrix for the system:
[1 1 1 | 1]
[1 1 -2 | -2]
[3 -2 1 | 6]
The elementary row operations we can use are:
1. Multiply a row by a nonzero constant.
2. Add or subtract a multiple of one row to another row.
3. Swap the position of two rows.
By examining the coefficients of the z terms in equations 2 and 3, we see that we can use the second elementary row operation to cancel the z term. Since the coefficient of z in equation 3 is 1 and the coefficient in equation 2 is -2, we can multiply equation 2 by 1 and equation 3 by 2, and then add equation 2 to equation 3.
Here are the steps to cancel the z term using elementary row operations:
1. Multiply equation 2 by 1:
R2' = R2 * 1
[1 1 -2 | -2] becomes [1 1 -2 | -2]
2. Multiply equation 3 by 2:
R3' = R3 * 2
[3 -2 1 | 6] becomes [6 -4 2 |12]
3. Add equation 2 (R2') to equation 3 (R3'):
R3' = R3' + R2'
[6 -4 2 | 12] + [1 1 -2 | -2] = [7 -3 0 | 10]
After performing these elementary row operations, the augmented matrix representing the system becomes:
[1 1 1 | 1]
[1 1 -2 | -2]
[7 -3 0 | 10]
Now, the z term is canceled from equation 2, and we can continue solving the system of equations using other elementary row operations or methods such as Gaussian elimination or matrix inversion.