Solving Linear Equations in Three Variables Using Rank Method :
Question 1 :
Test for consistency and if possible, solve the following systems of equations by rank method.
(iii) 2x + 2y + z = 5, x − y + z =1, 3x + y + 2z = 4
ρ (A) = 3, ρ ([A| B]) = 2
Since the ranks of matrices A and [A, B} are not equal, it has no solution.
(iv) 2x − y + z = 2, 6x − 3y + 3z = 6, 4x − 2y + 2z = 4
ρ (A) = 1, ρ ([A| B]) = 1
Since the ranks of matrices A and [A, B] are equal and it is less than 3, it has infinitely many solution.
2x - y + z = 2
Let y = s and z = t
2x - s + t = 2
2x = 2 + s - t
x = (2 + s - t)/2
Hence the solution is ((2 + s - t)/2, s, t) where s,t belongs to R.
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