Question

The system of equations
$
  ax + by + cz = q - r \\
  bx + cy + az = r - p \\
  cx + ay + bz = p - q \\
 $
Is
This question has multiple correct options
A. Consistent if $p = q = r$
B. Inconsistent if $a = b = c,$ and p, q, r are distinct
C. Consistent if a, b, c are distinct $a + b + c \ne 0$
D. None of these

Answer 1

Hint: System of equations means the system having two or more equations. It involves at most an x-variable, y-variable and the constant value. Here we are given a system of equations to check whether it is consistent or inconsistent. Here we will take the given options and its conditions to check.

Complete step by step answer:
If a consistent system of equations has exactly one solution then it is independent and if it has an infinite number of solutions then it is dependent whereas inconsistent systems of equations do not have any solution.First consider if $p = q = r$ then $x = 0,\;{\text{y = 0, z = 0}}$ will be the solution of the system of equations. Now, if a, b, c are distinct, then the determinant of the coefficient of the matrix is,
$\Rightarrow\dfrac{1}{2}(a + b + c)[{(b - c)^2} + {(c - a)^2} + {(a - b)^2}] \ne 0$.
Then the system of equations will have a unique solution.

Hence,the options A and C are the correct answer.

Note:Know the concepts of the consistent and inconsistent system of equations. Consistent system of equations has at least one solution whereas an inconsistent system of the equation has no solution. Observe the set of the given equations and determine the system.The solutions for the system of equations can be found using the elimination method or the combination of the elimination and the substitution method.