Given information: Two system of linear equations

(a) x−2y+z=−6 Equation 1

  y−5z=16 Equation 2

  z=−3 Equation 3

(b) x+y−2z=6 Equation 1

  y+3z=−8 Equation 2

  z=−3 Equation 3

Concept used:Gaussian Elimination is the process of using valid row operations on a matrix until it is in reduced row echelon form

Calculation:

The solution of the two system of linear equation is:

(a) x−2y+z=−6 Equation 1

  y−5z=16 Equation 2

  z=−3 Equation 3

From equation 3 the value of z is known. To solve for y , substitute z=−3 into equation 2 to obtain

  y−5(−3)=16y=16−15y=1

Finally, substitute y=1 and z=−3 into equation 1 to obtain

  x−2(1)+(−3)=−6x−2−3=−6x=−6+5x=−1

The solution of first system is x=−1,y=1,z=−3

Now, check solution of second system,

(b) x+y−2z=6 Equation 1

  y−5z=16 Equation 2

  z=−3 Equation 3

From equation 3 the value of z is known. To solve for y , substitute z=−3 into equation 2 to obtain

  y−5(−3)=16y=16−15y=1

Finally, substitute y=1 and z=−3 into equation 1 to obtain

  x+1−2(−3)=6x+1+6=6x=6−7x=−1

The solution of the second system is x=−1,y=1,z=−3

Therefore, two system of linear equations yield the same solution

The solution is x=−1,y=1,z=−3