Solve Using an Augmented Matrix x+7y=z+6 , x=2+y-z , x+y-3z=-4

Problem

{[x+7*y=z+6],[x=2+y−z],[x+y−3*z=−4])

Solution

1. Rearrange the equations into standard form a*x+b*y+c*z=d so that all variables are on the left and constants are on the right.

x+7*y−z=6

x−y+z=2

x+y−3*z=−4

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

[[1,7,−1,6],[1,−1,1,2],[1,1,−3,−4]]

3. Perform row operations to create zeros in the first column below the first row by calculating (R_2)−(R_1)⇒(R_2) and (R_3)−(R_1)⇒(R_3)

[[1,7,−1,6],[0,−8,2,−4],[0,−6,−2,−10]]

4. Simplify the second and third rows by dividing (R_2) by −2 and (R_3) by −2

[[1,7,−1,6],[0,4,−1,2],[0,3,1,5]]

5. Eliminate the y term in the third row by calculating 4*(R_3)−3*(R_2)⇒(R_3)

[[1,7,−1,6],[0,4,−1,2],[0,0,7,14]]

6. Solve for z using the third row.

7*z=14

z=2

7. Substitute z=2 into the second row equation to solve for y

4*y−2=2

4*y=4

y=1

8. Substitute y=1 and z=2 into the first row equation to solve for x

x+7*(1)−2=6

x+5=6

x=1

Final Answer

(x,y,z)=(1,1,2)

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