Solve Using a Matrix by Elimination x-y+4z=6 , 2x+y+z=1 , x+5y+z=-9

Problem

{[x−y+4*z=6],[2*x+y+z=1],[x+5*y+z=−9])

Solution

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

[[1,−1,4,6],[2,1,1,1],[1,5,1,−9]]

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

[[1,−1,4,6],[0,3,−7,−11],[0,6,−3,−15]]

3. Eliminate the y-term in the third row by performing the row operation (R_3)−2*(R_2)⇒(R_3)

[[1,−1,4,6],[0,3,−7,−11],[0,0,11,7]]

4. Solve for z using the third row of the matrix.

11*z=7

z=7/11

5. Solve for y by substituting z into the second row equation 3*y−7*z=−11

3*y−7*(7/11)=−11

3*y−49/11=−121/11

3*y=−72/11

y=−24/11

6. Solve for x by substituting y and z into the first row equation x−y+4*z=6

x−(−24/11)+4*(7/11)=6

x+24/11+28/11=66/11

x+52/11=66/11

x=14/11

Final Answer

(x,y,z)=(14/11,−24/11,7/11)

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