GATE CSE 2014 Set 1 | Question: 4

Question

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 ______________

Comments

Could we have done it like this:

step 1:

we first  find the solution of the first 3 equations.

step 2:

we then substitute the solution in the 4th equation to see iff it satisfies or not.

is this procedure correct?

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This is very short method

R4-> R4 + R1

Then R4 and R2 will become identical so

R4 -> R4 - R2

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@Doraemon  Yes you can do like that . Answers will be x= -11/5  y=19/5 and z=7/5

Answer 1

rank(Augmented Matrix) = rank(Matrix) = no of unknowns. Hence it has a unique solution

Comments

Don't you think that this might be a very lengthy solution? although it is the right way of getting answers for these kind of questions

Answer 2

It is having UNIQUE Solution .

 

Comments

In this question many people get trapped

Augmented matrix for given equations is:

3   4   1   1

2   0   1   -1

0   7   1   7

1   1   3   0

After converting into echelon form we obtain:

3   0   0   0

2   -8   0   0

0   21   45   0

1   -1   63   0

After seeing [00..|0] we think that there are infinitely many solutions. But the catch here is there is no free variable.

So there will be unique solution 

Method-2: 

By using rank

Rank[A] = 3 

Rank[A|b] = 3 

So rank=number of columns

There will be unique solution

Answer is 1