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+15 votes
1.7k views
Consider the following system of equations:  

$3x + 2y = 1 $

$4x + 7z = 1 $

$x + y + z = 3$

$x - 2y + 7z = 0$

The number of solutions for this system is ______________
asked in Linear Algebra by Veteran (106k points) | 1.7k views

4 Answers

+21 votes
Best answer
Since equation (2) - equation (1) produces equation (4), we have 3 independent equations in 3 variables, hence unique solution.

So answer is 1.
answered by Veteran (11k points)
selected
I could not understand this.. Can u please elaborate more
And solution is x=13,y=-19,z=9 only single solution
Finding rank of 4x4 matrix is quite time consuming .But logic should be clear.

Which three equation are used to determine the answers is not mentioned.
+11 votes

sorry for my handwriting!

answered by Active (2.5k points)
how did you make R4 all zeros?
if R4 is all zeros , then there is 3x3 square  matrix inside  the augmented matrix , for which the determinant is zero .. hence rank of the augmented matrix cannot be 3.
@shaurya

rank of augmented matrix(AB) >= rank of A.

In the pic R= R4 - R2 ..... And To be unique solution R(A|B) = R(A) =  n where n is # of variables ...

0 votes
Can someone plz find the rank of thie matrix using row transformations.I m not able to do so.Plz help
answered by Active (1.2k points)
rank(A) = rank(AB) = n (no. of unknowns) =3
Even i am getting rank as 4 although its not possible.
Rank will be 4 if you solve all 4 equations together. But note that 2 rows in Echelon form will be identical.
0 votes
rank(Augmented Matrix) = rank(Matrix) = no of unknowns. Hence it has a unique solution
answered by Boss (7.8k points)


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