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 8.3k 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 31 votes 31 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 Ayush Upadhyaya commented Jul 15, 2018 reply Follow Share Number of free variables=n-r=2-2=0, so no infinite, only unique solutions. 10 votes 10 votes Verma Ashish commented Feb 4, 2019 reply Follow Share I think for any set of linear non homogeneous equations two distinct solutions (option B) can never be possible.. 2 votes 2 votes Lakshman Bhaiya commented Nov 30, 2019 reply Follow Share @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. 0 votes 0 votes ankitgupta.1729 commented Nov 30, 2019 reply Follow Share @Lakshman Patel RJIT How is it possible that a linear non-homogeneous system of simultaneous equations have 'only' 2 distinct solutions ? 3 votes 3 votes Lakshman Bhaiya commented Nov 30, 2019 i edited by Lakshman Bhaiya Nov 30, 2019 reply Follow Share @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. Non-trivial solution (infinite many solution) Trivial solution (unique solution) 5 votes 5 votes ankitgupta.1729 commented Nov 30, 2019 reply Follow Share 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 :) 7 votes 7 votes Aalok8523 commented May 26, 2020 reply Follow Share @ankitgupta.1729 why "When at least 2 of the lines are parallel then it will give no solution" ? why not unique solution is possible. 0 votes 0 votes ankitgupta.1729 commented Jun 21, 2020 reply Follow Share @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. 1 votes 1 votes 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.
3 votes 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 :) 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.