Solve Using a Matrix by Elimination 2x-8y-z=-23 , x+9y-9z=-36 , 6x+y+z=67

Problem

{[2*x−8*y−z=−23],[x+9*y−9*z=−36],[6*x+y+z=67])

Solution

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

[[2,−8,−1,|,−23],[1,9,−9,|,−36],[6,1,1,|,67]]

2. Swap the first and second rows ((R_1)↔(R_2) to get a 1 in the top-left corner.

[[1,9,−9,|,−36],[2,−8,−1,|,−23],[6,1,1,|,67]]

3. Eliminate the xterms in the second and third rows using (R_2)−2*(R_1)→(R_2) and (R_3)−6*(R_1)→(R_3)

[[1,9,−9,|,−36],[0,−26,17,|,49],[0,−53,55,|,283]]

4. Divide the second row by −26 to create a leading 1 (−1/26*(R_2)→(R_2).

[[1,9,−9,|,−36],[0,1,−17/26,|,−49/26],[0,−53,55,|,283]]

5. Eliminate the yterm in the third row using (R_3)+53*(R_2)→(R_3)

[[1,9,−9,|,−36],[0,1,−17/26,|,−49/26],[0,0,529/26,|,4761/26]]

6. Solve for z by multiplying the third row by 26/529

529/26*z=4761/26

z=9

7. Substitute z=9 into the second row equation to solve for y

y−17/26*(9)=−49/26

y=153/26−49/26

y=104/26

y=4

8. Substitute y=4 and z=9 into the first row equation to solve for x

x+9*(4)−9*(9)=−36

x+36−81=−36

x−45=−36

x=9

Final Answer

(x,y,z)=(9,4,9)

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