Question

Classify the system and identify the number of solutions.

x − 2y − 2z = 8
2x + 6y − 2z = 9
8x − 6y − 14z = 11


(a) Consistent, independent; one solution
(b) Consistent, dependent; infinite solutions
(c) Inconsistent; no solutions
(d) None of the above

Answer 1

The solution of equations is identified to be as: Consistent, dependent; infinite solutions. The answer is option B.

Step-by-step explanation:

To classify the system of equations and identify the number of solutions, we can use elimination or Gaussian elimination.

Steps to solve:

Convert the system of equations to an augmented matrix:

[1 -2 -2 | 8]

[2 6 -2 | 9]

[8 -6 -14 | 11]

Perform Gaussian elimination to eliminate the leading term in the second and third rows:

[1 -2 -2 | 8]

[0 10 0 | 1]

[0 2 -10 | 3]

Back-solve to find the values of z, y, and x:

z = -3/10

y = 1/10

x = -2/5 + 4z = 1/5

Determine the number of solutions: Since the variables are not unique, the system is consistent and dependent, with infinite solutions.

Therefore, the answer is: Consistent, dependent; infinite solutions. Option B is answer.