Solve Using a Matrix by Elimination -3x-5y+z=3 , 5x+y-6z=2 , x+4y+z=-6

Problem

{[−3*x−5*y+z=3],[5*x+y−6*z=2],[x+4*y+z=−6])

Solution

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

[[−3,−5,1,|,3],[5,1,−6,|,2],[1,4,1,|,−6]]

2. Swap the first and third rows to get a 1 in the top-left position (pivot).

[[1,4,1,|,−6],[5,1,−6,|,2],[−3,−5,1,|,3]]

3. Eliminate the x-terms in the second and third rows using the first row.

(R_2)←(R_2)−5*(R_1)

(R_3)←(R_3)+3*(R_1)

[[1,4,1,|,−6],[0,−19,−11,|,32],[0,7,4,|,−15]]

4. Eliminate the y-term in the third row using the second row.

(R_3)←19*(R_3)+7*(R_2)

[[1,4,1,|,−6],[0,−19,−11,|,32],[0,0,−1,|,−61]]

5. Solve for z using the third row.

−z=−61

z=61

6. Back-substitute z into the second row to solve for y

−19*y−11*(61)=32

−19*y−671=32

−19*y=703

y=−37

7. Back-substitute y and z into the first row to solve for x

x+4*(−37)+61=−6

x−148+61=−6

x−87=−6

x=81

Final Answer

(x,y,z)=(81,−37,61)

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