Solve Using a Matrix by Elimination x-3y+3z=-6 , 4x+y-z=-11 , 3x+4y-5z=-8

Problem

{[x−3*y+3*z=−6],[4*x+y−z=−11],[3*x+4*y−5*z=−8])

Solution

1. Write the augmented matrix representing the system of linear equations.

[[1,−3,3,−6],[4,1,−1,−11],[3,4,−5,−8]]

2. Eliminate the x-terms in the second and third rows by performing row operations (R_2)−4*(R_1)→(R_2) and (R_3)−3*(R_1)→(R_3)

[[1,−3,3,−6],[0,13,−13,13],[0,13,−14,10]]

3. Simplify the second row by dividing by 13 using the operation 1/13*(R_2)→(R_2)

[[1,−3,3,−6],[0,1,−1,1],[0,13,−14,10]]

4. Eliminate the y-term in the third row by performing the operation (R_3)−13*(R_2)→(R_3)

[[1,−3,3,−6],[0,1,−1,1],[0,0,−1,−3]]

5. Solve for z by multiplying the third row by −1

[[1,−3,3,−6],[0,1,−1,1],[0,0,1,3]]

z=3

6. Back-substitute the value of z into the second row equation to find y

y−z=1

y−3=1

y=4

7. Back-substitute the values of y and z into the first row equation to find x

x−3*y+3*z=−6

x−3*(4)+3*(3)=−6

x−12+9=−6

x−3=−6

x=−3

Final Answer

(x,y,z)=(−3,4,3)

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