SOLUTION: The augmented matrix is in row echelon form and represents a linear system. Solve the system by using backward substitution, if possible. Write the solution as either an ordered pa

Algebra ->  Matrices-and-determiminant -> SOLUTION: The augmented matrix is in row echelon form and represents a linear system. Solve the system by using backward substitution, if possible. Write the solution as either an ordered pa      Log On


   

Question 440213: The augmented matrix is in row echelon form and represents a linear system. Solve the system by using backward substitution, if possible. Write the solution as either an ordered pair or ordered triple.
1 -1 2 8
0 1 -4 2
0 0 0 0
Answer by Edwin McCravy(20065) About Me  (Show Source):
You can put this solution on YOUR website!


This matrix



represents this system of equations:

system%281x-1y%2B2z=8%2C%0D%0A0x%2B1y-4z=2%2C%0D%0A0x%2B0y%2B0z=0%29

or written simpler:

system%28x-y%2B2z=8%2C%0D%0Ay-4z=2%2C%0D%0A0z=0%29

The bottom equation

               0z = 0

has ANY real number as a solution.
Suppose the number k represents ANY
solution to 0z = 0, then  

                z = k

Substitute that into 

           y - 4z = 2

           y - 4k = 2

                y = 2 + 4k

Substitute z = k and y = 2 + 4k into

       x - y + 2z = 8

x - (2 + 4k) + 2k = 8

  x - 2 - 4k + 2k = 8

       x - 2 - 2k = 8

                x = 10 + 2k


So the solution is this ordered triple:

(x, y, z) = (10 + 2k, 2 + 4k, k)