Solve Using a Matrix by Elimination -x-2y-2z=-3 , -x-5y-2z=-9 , -x-5y-5z=-6

Problem

{[−x−2*y−2*z=−3],[−x−5*y−2*z=−9],[−x−5*y−5*z=−6])

Solution

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

[[−1,−2,−2,|,−3],[−1,−5,−2,|,−9],[−1,−5,−5,|,−6]]

2. Eliminate the x terms in the second and third rows by subtracting the first row from them.

(R_2)−(R_1)→(R_2)

(R_3)−(R_1)→(R_3)

[[−1,−2,−2,|,−3],[0,−3,0,|,−6],[0,−3,−3,|,−3]]

3. Eliminate the y term in the third row by subtracting the second row from the third row.

(R_3)−(R_2)→(R_3)

[[−1,−2,−2,|,−3],[0,−3,0,|,−6],[0,0,−3,|,3]]

4. Solve for z using the third row.

−3*z=3

z=−1

5. Solve for y using the second row.

−3*y=−6

y=2

6. Solve for x by substituting y and z into the first row equation.

−x−2*(2)−2*(−1)=−3

−x−4+2=−3

−x−2=−3

−x=−1

x=1

Final Answer

(x,y,z)=(1,2,−1)

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