28 votes 28 votes How many solutions does the following system of linear equations have? $-x + 5y = -1$ $x - y = 2$ $x + 3y = 3$ infinitely many two distinct solutions unique none Linear Algebra gatecse-2004 linear-algebra system-of-equations normal + – Kathleen asked Sep 18, 2014 Kathleen 8.4k views answer comment Share Follow See all 2 Comments See all 2 2 Comments reply its_vv commented Nov 27, 2022 reply Follow Share 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? 0 votes 0 votes parth023 commented Apr 5, 2023 reply Follow Share that will create [ 0 0 | nonzero ] situation and hence no solution. 0 votes 0 votes Please log in or register to add a comment.
Best answer 32 votes 32 votes answer = C rank = r(A) = r(A|B) = 2 rank = total number of variables Hence, unique solution amarVashishth answered Jan 13, 2016 • selected Nov 30, 2016 by focus _GATE amarVashishth comment Share Follow See all 11 Comments See all 11 11 Comments reply Show 8 previous comments Aalok8523 commented Jun 21, 2020 reply Follow Share Thanks for reply bro. 1 votes 1 votes yuyutsu commented Aug 19, 2022 reply Follow Share So b is out of question right? Coz here we have linear functions. If meeting exists only at one point. 0 votes 0 votes Argharupa Adhikary commented Sep 17, 2022 reply Follow Share NICE 1 votes 1 votes Please log in or register to add a comment.
23 votes 23 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 Digvijay Pandey answered May 4, 2015 Digvijay Pandey comment Share Follow See all 2 Comments See all 2 2 Comments reply mrinmoyh commented May 28, 2019 reply Follow Share @Digvijay Pandey sir, for given this type of equation i.e non-homogeneous eqn, how can I distinguish b/w infinitely many & no solution. 0 votes 0 votes kp_12 commented Sep 13, 2019 reply Follow Share 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 0 votes 0 votes Please log in or register to add a comment.
4 votes 4 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 :) gatesjt answered Nov 8, 2016 gatesjt comment Share Follow See all 4 Comments See all 4 4 Comments reply HeadShot commented Nov 8, 2018 i moved by Lakshman Bhaiya Nov 30, 2019 reply Follow Share $2X2$ Minor with Determinant non - zero. 0 votes 0 votes mrinmoyh commented May 28, 2019 reply Follow Share 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) 2 votes 2 votes Satbir commented Oct 25, 2019 reply Follow Share Yes rank is coming 2 $\implies$ we have only 2 independent equations not 3. 0 votes 0 votes amit166 commented Nov 9, 2019 i moved by Lakshman Bhaiya Nov 30, 2019 reply Follow Share All three line are intersect at only point X=9/4 and y=1/4 so unique solution exist 1 votes 1 votes Please log in or register to add a comment.
0 votes 0 votes rank[A] = 2 and rank[A|B] = 2 it is unique solution . Rahul_kumar3 answered Nov 25, 2023 Rahul_kumar3 comment Share Follow See 1 comment See all 1 1 comment reply SASIDHAR_1 commented Feb 25 reply Follow Share In this question many people get trapped.Method-1:For given equations Augmented matrix[A|b] is : -1 1 55 -1 3-1 2 3R2→ R2+R1R3→R3+R1 ThenR3→R3-2R2After converting into echelon form we obtain:-1 0 05 4 0-1 1 0After 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 solutionMethod-2:The matrix is 3X2 matrixFind rank[A] and rank[A|b] rank[A] = 2 rank[A|b]=2 So number of free columns = 0So there will be unique solutiontherefore the option-C is correct 0 votes 0 votes Please log in or register to add a comment.