Solve Using a Matrix by Elimination x-5y-z=10 , -x+7y-z=18 , 5x-4y-2z=-14

Problem

{[x−5*y−z=10],[−x+7*y−z=18],[5*x−4*y−2*z=−14])

Solution

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

[[1,−5,−1,|,10],[−1,7,−1,|,18],[5,−4,−2,|,−14]]

2. Eliminate the x-terms in the second and third rows by performing row operations (R_2)+(R_1)→(R_2) and (R_3)−5*(R_1)→(R_3)

[[1,−5,−1,|,10],[0,2,−2,|,28],[0,21,3,|,−64]]

3. Simplify the second row by dividing by 2 ((R_2)/2→(R_2) to create a leading 1.

[[1,−5,−1,|,10],[0,1,−1,|,14],[0,21,3,|,−64]]

4. Eliminate the y-term in the third row by performing (R_3)−21*(R_2)→(R_3)

[[1,−5,−1,|,10],[0,1,−1,|,14],[0,0,24,|,−358]]

5. Solve for z using the third row equation 24*z=−358

z=−358/24

z=−179/12

6. Solve for y using back-substitution into the second row equation y−z=14

y−(−179/12)=14

y=168/12−179/12

y=−11/12

7. Solve for x using back-substitution into the first row equation x−5*y−z=10

x−5*(−11/12)−(−179/12)=10

x+55/12+179/12=120/12

x+234/12=120/12

x=−114/12

x=−19/2

Final Answer

(x,y,z)=(−19/2,−11/12,−179/12)

Want more problems? Check here!