36 votes 36 votes 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 ______________ Linear Algebra gatecse-2014-set1 linear-algebra system-of-equations numerical-answers normal + – go_editor asked Sep 26, 2014 go_editor 13.2k views answer comment Share Follow See all 3 Comments See all 3 3 Comments reply Doraemon commented Jun 2, 2019 reply Follow Share 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? 1 votes 1 votes shubham.csec commented Dec 31, 2019 reply Follow Share This is very short method R4-> R4 + R1 Then R4 and R2 will become identical so R4 -> R4 - R2 2 votes 2 votes aforgate commented Jan 4, 2021 reply Follow Share @Doraemon Yes you can do like that . Answers will be x= -11/5 y=19/5 and z=7/5 1 votes 1 votes Please log in or register to add a comment.
Best answer 45 votes 45 votes Since, equation $(2)$ - equation $(1)$ produces equation $(4)$, we have $3$ independent equations in $3$ variables, hence unique solution. So, answer is $1.$ Happy Mittal answered Sep 28, 2014 • edited Jun 8, 2018 by Milicevic3306 Happy Mittal comment Share Follow See all 3 Comments See all 3 3 Comments reply anchitjindal07 commented Oct 2, 2017 reply Follow Share I could not understand this.. Can u please elaborate more 0 votes 0 votes saxena0612 commented Oct 11, 2017 reply Follow Share And solution is x=13,y=-19,z=9 only single solution 0 votes 0 votes Sona Barman commented Dec 20, 2017 i edited by Sona Barman Dec 20, 2017 reply Follow Share Finding rank of 4x4 matrix is quite time consuming .But logic should be clear. Which three equation are used to determine the answers is not mentioned. 0 votes 0 votes Please log in or register to add a comment.
49 votes 49 votes sorry for my handwriting! swap_it answered Jan 15, 2017 swap_it comment Share Follow See all 4 Comments See all 4 4 Comments reply shaurya vardhan commented Jan 1, 2018 reply Follow Share how did you make R4 all zeros? 1 votes 1 votes shaurya vardhan commented Jan 1, 2018 reply Follow Share if R4 is all zeros , then there is 3x3 square matrix inside the augmented matrix , for which the determinant is zero .. hence rank of the augmented matrix cannot be 3. 1 votes 1 votes vishalshrm539 commented Jan 14, 2018 reply Follow Share @shaurya rank of augmented matrix(AB) >= rank of A. 0 votes 0 votes Puja Mishra commented Jan 23, 2018 reply Follow Share In the pic R4 = R4 - R2 ..... And To be unique solution R(A|B) = R(A) = n where n is # of variables ... 0 votes 0 votes Please log in or register to add a comment.
16 votes 16 votes Add first two equations and you will get $7x+2y+7z=2$ remaining equations are $x+y+z=3$ and $x-2y+7z=0$ My augmented matrix $\left[\begin{array}{ccc|c} 1&1&1&3 \\ 7&2&7&2 \\ 1&-2&7&0 \\ \end{array} \right ]$ do $R_2-7R_1 \rightarrow R_2$ and $R_3-R_1 \rightarrow R_3$ $\left[\begin{array}{ccc|c} 1&1&1&3 \\ 0&-5&0&-19 \\ 0&-3&6&-3 \\ \end{array} \right ]$ do $5R_3 - 3R_2 \rightarrow R_3$ $\left[\begin{array}{ccc|c} 1&1&1&3 \\ 0&-5&0&-19 \\ 0&0&30&42 \\ \end{array} \right ]$ your Augmented matrix has same rank as the coefficient matrix=3=number of unknowns, so only 1 solution possible. Ayush Upadhyaya answered Jul 15, 2018 Ayush Upadhyaya comment Share Follow See all 4 Comments See all 4 4 Comments reply Satbir commented Dec 26, 2018 reply Follow Share should we first always reduce the equations to no. of variables in equation and then solve augmented matrix ? 0 votes 0 votes mrinmoyh commented May 29, 2019 reply Follow Share @Ayush Upadhyaya Is it fruitful or can I use it for general method as satbir said in his comment. Can it harm solution for any example?? 0 votes 0 votes Sambhrant Maurya commented Oct 5, 2019 reply Follow Share @Satbir Does an overdetermined system(No of equations> no of variables) always have a unique solution? 0 votes 0 votes Satbir commented Oct 5, 2019 reply Follow Share NO. x + y =2 2x+2y =4 6x + 6y = 12 These will have infinite solution. 3 votes 3 votes Please log in or register to add a comment.
0 votes 0 votes Can someone plz find the rank of thie matrix using row transformations.I m not able to do so.Plz help Gate Mm answered Dec 2, 2015 Gate Mm comment Share Follow See all 3 Comments See all 3 3 Comments reply Nit9 commented Dec 26, 2015 reply Follow Share rank(A) = rank(AB) = n (no. of unknowns) =3 3 votes 3 votes Adiaspirant commented Dec 25, 2016 reply Follow Share Even i am getting rank as 4 although its not possible. 0 votes 0 votes Ankush Kundaliya commented Jan 13, 2017 reply Follow Share Rank will be 4 if you solve all 4 equations together. But note that 2 rows in Echelon form will be identical. 0 votes 0 votes Please log in or register to add a comment.