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

Problem

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

Solution

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

[[1,2,−1,|,−3],[2,−4,1,|,−7],[−2,2,−3,|,4]]

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

[[1,2,−1,|,−3],[0,−8,3,|,−1],[0,6,−5,|,−2]]

3. Create a leading 1 in the second row by dividing (R_2) by −8

[[1,2,−1,|,−3],[0,1,−3/8,|,1/8],[0,6,−5,|,−2]]

4. Eliminate the y-term in the third row using the operation (R_3)−6*(R_2)⇒(R_3)

[[1,2,−1,|,−3],[0,1,−3/8,|,1/8],[0,0,−11/4,|,−11/4]]

5. Solve for z by dividing the third row by −11/4

[[1,2,−1,|,−3],[0,1,−3/8,|,1/8],[0,0,1,|,1]]

6. Use back-substitution to find y using the second row equation y−3/8*z=1/8

y−3/8*(1)=1/8

y=4/8=1/2

7. Use back-substitution to find x using the first row equation x+2*y−z=−3

x+2*(1/2)−1=−3

x+1−1=−3

x=−3

Final Answer

(x,y,z)=(−3,1/2,1)

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