Solve Using a Matrix by Elimination 4x-y+3z=-8 , x-8y+5z=3 , 2y-7z=-4

Problem

{[4*x−y+3*z=−8],[x−8*y+5*z=3],[2*y−7*z=−4])

Solution

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

[[4,−1,3,|,−8],[1,−8,5,|,3],[0,2,−7,|,−4]]

2. Swap rows to place a 1 in the first pivot position (row 1, column 1).

(R_1)↔(R_2)

[[1,−8,5,|,3],[4,−1,3,|,−8],[0,2,−7,|,−4]]

3. Eliminate the x-term in the second row by creating a zero in the first column.

(R_2)−4*(R_1)→(R_2)

[[1,−8,5,|,3],[0,31,−17,|,−20],[0,2,−7,|,−4]]

4. Swap rows to make the arithmetic easier for the next elimination step.

(R_2)↔(R_3)

[[1,−8,5,|,3],[0,2,−7,|,−4],[0,31,−17,|,−20]]

5. Eliminate the y-term in the third row to reach row-echelon form.

(R_3)−31/2*(R_2)→(R_3)

[[1,−8,5,|,3],[0,2,−7,|,−4],[0,0,183/2,|,42]]

6. Solve for z using the third row.

183/2*z=42

z=84/183

z=28/61

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

2*y−7*(28/61)=−4

2*y−196/61=−244/61

2*y=−48/61

y=−24/61

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

x−8*(−24/61)+5*(28/61)=3

x+192/61+140/61=183/61

x+332/61=183/61

x=−149/61

Final Answer

(x,y,z)=(−149/61,−24/61,28/61)

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