Solve Using a Matrix by Elimination x+y+2z=-1 , x+3y-6z=5 , 2x+5y-8z=7

Problem

{[x+y+2*z=−1],[x+3*y−6*z=5],[2*x+5*y−8*z=7])

Solution

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

[[1,1,2,|,−1],[1,3,−6,|,5],[2,5,−8,|,7]]

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

[[1,1,2,|,−1],[0,2,−8,|,6],[0,3,−12,|,9]]

3. Simplify the second row by dividing all elements by 2 ((R_2)/2→(R_2).

[[1,1,2,|,−1],[0,1,−4,|,3],[0,3,−12,|,9]]

4. Eliminate the y-term from the third row by performing the operation (R_3)−3*(R_2)→(R_3)

[[1,1,2,|,−1],[0,1,−4,|,3],[0,0,0,|,0]]

5. Identify the nature of the system based on the resulting row-echelon form. The third row 0=0 indicates that the system is dependent and has infinitely many solutions.

6. Express the variables in terms of a parameter. Let z=t From the second row, y−4*z=3 so y=3+4*t From the first row, x+y+2*z=−1

x+(3+4*t)+2*t=−1

x+3+6*t=−1

x=−4−6*t

Final Answer

(x,y,z)=(−4−6*t,3+4*t,t)* for any real number *t

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