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

  • $-x + 5y = -1$
  • $x - y = 2$
  • $x + 3y = 3$
  1. infinitely many
  2. two distinct solutions
  3. unique
  4. none
in Linear Algebra by Veteran (52.2k points) | 1.9k views

3 Answers

+25 votes
Best answer
answer = C

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

rank = total number of variables
Hence, unique solution
by Boss (30.8k points)
selected by
+5
Number of free variables=n-r=2-2=0, so no infinite, only unique solutions.
+1

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

 

0

@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.

+2

@Lakshman Patel RJIT

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

+1

@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)

 

+1
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 :)
+18 votes
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
by Veteran (60.9k points)
0

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

0

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

+3 votes

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 :)

by Junior (905 points)
0

$2X2$  Minor with Determinant non - zero.

0
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)
0
Yes rank is coming 2 $\implies$ we have only 2 independent equations not 3.
+1
All three line are intersect at only point X=9/4 and y=1/4 so unique solution exist
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