Solve Using a Matrix by Elimination x+y+z=-4 , x-y+4z=6 , 4x+y+z=-16

Problem

{[x+y+z=−4],[x−y+4*z=6],[4*x+y+z=−16])

Solution

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

[[1,1,1,|,−4],[1,−1,4,|,6],[4,1,1,|,−16]]

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

[[1,1,1,|,−4],[0,−2,3,|,10],[0,−3,−3,|,0]]

3. Simplify the third row by dividing by −3 to make the calculations easier (−1/3*(R_3)→(R_3).

[[1,1,1,|,−4],[0,−2,3,|,10],[0,1,1,|,0]]

4. Swap the second and third rows to place a 1 in the second row's pivot position ((R_2)↔(R_3).

[[1,1,1,|,−4],[0,1,1,|,0],[0,−2,3,|,10]]

5. Eliminate the y-term from the third row by performing (R_3)+2*(R_2)→(R_3)

[[1,1,1,|,−4],[0,1,1,|,0],[0,0,5,|,10]]

6. Solve for z using the third row.

5*z=10

z=2

7. Solve for y using back-substitution into the second row.

y+z=0

y+2=0

y=−2

8. Solve for x using back-substitution into the first row.

x+y+z=−4

x−2+2=−4

x=−4

Final Answer

(x,y,z)=(−4,−2,2)

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