Solve Using a Matrix by Elimination 2x-12y-4z=5 , x+7y-z=-4 , x+3y=7

Problem

{[2*x−12*y−4*z=5],[x+7*y−z=−4],[x+3*y=7])

Solution

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

[[2,−12,−4,|,5],[1,7,−1,|,−4],[1,3,0,|,7]]

2. Swap rows to place a 1 in the first pivot position ((R_1)↔(R_2).

[[1,7,−1,|,−4],[2,−12,−4,|,5],[1,3,0,|,7]]

3. Eliminate the first column entries below the pivot by performing (R_2)−2*(R_1)→(R_2) and (R_3)−(R_1)→(R_3)

[[1,7,−1,|,−4],[0,−26,−2,|,13],[0,−4,1,|,11]]

4. Swap rows to simplify the second pivot ((R_2)↔(R_3).

[[1,7,−1,|,−4],[0,−4,1,|,11],[0,−26,−2,|,13]]

5. Eliminate the second column entry below the pivot by performing (R_3)−6.5*(R_2)→(R_3)

[[1,7,−1,|,−4],[0,−4,1,|,11],[0,0,−8.5,|,−58.5]]

6. Solve for z using the third row.

−8.5*z=−58.5

z=6.9

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

−4*y+6.9=11

−4*y=4.1

y=−1.025

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

x+7*(−1.025)−6.9=−4

x−7.175−6.9=−4

x−14.075=−4

x=10.075

Final Answer

(x,y,z)=(10.075,−1.025,6.9)

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