GATE CSE 2004 | Question: 71

Question

How many solutions does the following system of linear equations have?

• $-x + 5y = -1$
• $x - y = 2$
• $x + 3y = 3$

• A. infinitely many
• B. two distinct solutions
• C. unique
• D. none

Comments

No. of variables=2

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Ans

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$Concept:$

Answer 1

answer = C

rank = r(A) = r(A|B) = 2

rank = total number of variables
Hence, unique solution

Comments

Number of free variables=n-r=2-2=0, so no infinite, only unique solutions.

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I think for any set of linear non homogeneous equations two distinct solutions (option B) can never be possible..

 

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@Verma Ashish

I think for any set of linear non homogeneous equations two distinct solutions (option B) can never be possible..

Your statement true for linear homogeneous equations, not for linear non-homogeneous equations.

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@Lakshman Patel RJIT

How is it possible that a linear non-homogeneous system of simultaneous equations have 'only' 2 distinct solutions ?

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@ankitgupta.1729

see my comment carefully, I say for linear homogeneous case.

$$\mathbf{AX = 0}$$ is called linear homogeneous, and it has only two cases because it is a consistent system.

1. Non-trivial solution (infinite many solution)
2. Trivial solution (unique solution)

 

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Yes, it is correct.

We can also solve this question without knowing rank, free variables,matrix , determinant etc...

We are given 3 lines.

Now , when these 3 lines lie on each other then it will give infinitely many solutions.

When at least 2 of the lines are parallel then it will give no solution.

Otherwise we will get unique solution.

Now, 3 lines will lie on each other when all 3 lines are same like x+y = 1 and 2x + 2y =2 and 3x + 3y = 3.

Now, most important thing is when 2 lines are parallel or lie on each then both lines must have the same slope or vice versa.

So, when we have a  set of lines and if they all have different slopes then neither they can be parallel nor they can lie on each other.

In other words, when set of lines have different solpes then we can't get infinitely many solution or no solution.

Since , here all three lines have different slopes , so we can't get  the case of infinitely many solutions or no solution.

So, by just seeing the slope, we can solve this question very quickly in case of given non-homogeneous system :)

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@ankitgupta.1729

why "When at least 2 of the lines are parallel then it will give no solution" ? why not unique solution is possible.

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@Aalok8523

unique solution means a common point which satisfies all the 3 equations. You will get the unique solution when all 3 lines intersect at a common point.  If you have at least 2 parallel lines then you will not get the common point between these 2 lines. Hence, you will not get the unique solution.

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Thanks for reply bro.

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So b is out of question right? Coz here we have linear functions. If meeting exists only at one point.

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NICE

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this is simplest answer given and anyways to get the solution we must have to use gaussian elimination method to get the rank and then we can say number of solution possible.

I was confused while solving because two rows are getting pivot elements and the last row is having [ 0 0 |non-zero ] and i've thought that if such a situtation occurs then it will defintely have no solution.So I've chose option D but actual answer is option C because already we've got two pivot elements that means ,given two cols are linear independent and so rank(A) =2 = two variables . And evenun augmented matrix the last row is not a pivot element to be considered.so it has unique solution

Answer 2

C unique solution..
3 equation , 2 variable.
solve any two equation and check 3rd equation by putting values in 3rd equation.
x = 9/4 , y = 1/4

Comments

@Digvijay Pandey sir, for given this type of equation i.e non-homogeneous eqn, how can I distinguish b/w infinitely many & no solution.

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When rank(A) != rank(AB) ==> No Solution since (AX=B ) is inconsistent.

When [ rank(A) = rank(AB) ] < n (no of unknown variables) ==> Infinitely many Solution

Answer 3

DIFFERENT APPROACH

all the equations are linearly independent. i.e non of the equations can be obtained by multiplying one of equation with a number (this means that no two vectors overlap each other leading to a rank of 3). therefore there will be a unique solution. 

Correct me if wrong :)

Comments

$2X2$  Minor with Determinant non - zero.

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how all equations are linearly independent.

multiplying -2 with eqn no. 2 & then add with eqn. no. 3 we'll get eqn no. 1

-2*(eqn2)+(eqn3) = (eqn1)

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Yes rank is coming 2 $\implies$ we have only 2 independent equations not 3.

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All three line are intersect at only point X=9/4 and y=1/4 so unique solution exist

Answer 4

Determinant exists for square matrix only , so if we try other methods we will find that the matrix has unique solution

Answer 5

Simple Question, correct answer is option-C 

Answer 6

rank[A] = 2 and rank[A|B] = 2

it is unique solution .

Comments

In this question many people get trapped.

Method-1:

For given equations Augmented matrix[A|b] is  :

-1   1   5

5   -1   3

-1   2   3

R2→ R2+R1

R3→R3+R1 

Then

R3→R3-2R2

After converting into echelon form we obtain:

-1   0   0

5   4   0

-1   1    0

After seeing [00…0] we think that infinite solution is the correct answer 

But the catch here is for infinite solution there must me atleast one free column. But in this case there is no free column so the answer here is unique solution

Method-2:

The matrix is 3X2 matrix

Find rank[A] and rank[A|b] 

rank[A] = 2 

rank[A|b]=2 

So number of free columns = 0

So there will be unique solution

therefore the option-C is correct