Question

Solve the system of equations using matrices. Use Gaussian elimination with back-substitution.

a)x=1,y=2,z=2
b)x=2,y=1,z=2
c)x=3,y=−1,z=4
d)x=−2,y=3,z=−1
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Answer 1

To solve the system of equations using matrices and Gaussian elimination with back-substitution, rewrite the system of equations in matrix form and perform row operations to eliminate coefficients below the main diagonal. Use back-substitution to solve for the variables.

Step-by-step explanation:

To solve the system of equations using matrices and Gaussian elimination with back-substitution, we need to rewrite the system of equations in matrix form. The system of equations can be represented as:

[1 1 -1 | 2]
[2 5 -1 | 3]
[4 -1 1 | 1]

Using row operations, we can eliminate the coefficients below the main diagonal and obtain an upper triangular system of equations:

[1 1 -1 | 2]
[0 3 1 | -1]
[0 0 4 | 2]

Finally, we can use back-substitution to solve for the variables. Starting from the last equation, we can find the value of z, then substitute it back into the second equation to find y, and finally substitute the values of y and z into the first equation to find x.