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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

### 1 comment

What will be the ans in case of rank is Greater than the number of variables in the matrix?

-x +5y = -1

3x -7y = 2

X + 8 y = 3

In this case rank (3) > number of variables (2)  .

Now what will be the ans?

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

rank = total number of variables
Hence, unique solution

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

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

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

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

moved

$2X2$  Minor with Determinant non - zero.

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