TANCET 2016 System of Equations

Question

Comments

$2)K=3$

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Question is not shown, please edit the question.

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Lakshman Patel RJIT It was mistakenly deleted while adding the tag to the question

Answer 1

Given that the system of equations is $:x+y+z=6$----->(1)

                                                                     $x+2y+3z=10$----->(2)

                                                                      $x+2y+kz=5$----->(3)

Write in terms of matrix$:$$A=\begin{bmatrix} 1 &1 &1 \\ 1 &2 &3 \\ 1 &2 &k \end{bmatrix}$ and   $B=\begin{bmatrix} 6\\10 \\5 \end{bmatrix}$

Now we can write in the form of Augmented matrix $[A:B]=\begin{bmatrix} 1 &1 & 1:6\\ 1 &2 &3 :10\\ 1 &2 &k :5 \end{bmatrix}$

Perform the some operation in the row $3$.

                                   $R_{3} \rightarrow R_{3}-R_{2}$

we get,$[A:B]$$=\begin{bmatrix} 1 &1 & 1:6 \\ 1 &2 &3 :10\\ 0 &0 &k-3 :-5 \end{bmatrix}$

$Case(1):$ $For$  $No$  $Solution$

$If$ $Rank(A)\neq Rank([A:B])$

we can simply put $k-3=0$,So we get $Rank(A)=2$ $and$ $Rank([A:B])=3$, which is $2\neq3$

So$,k=3$

$Case(2):$ $Unique$ $Solution$

Rank(A)=Rank([A:B])=Unknowns(Number of Variables)

So,we can put $k-3\neq0$,which give Rank(A)=Rank([A:B])=Unknowns=3.

So$, k\neq3$

$Case(3):$ $Infinitely$  $many$  $numbers$  $of$ $solutions$

Rank(A)=Rank([A:B])<Unknowns(Number of Variables)

Which is not possible for this question.

Comments

yaa i got the same :D

Answer 2

One doesn't need to apply Gaussian Elimination for this question. It can be solved by simple inspection.

First of all one need to understand, a linear equation in N independent variables represent an N-1 dimensional hyperplane in N dimensional space. So, a linear equation in 3 variables represent a 2D plane in 3D space.

Now let's observe the 3 equations given :

x + y + z = 6

x + 2y + 3z = 10 and x + 2y + kz = 5.

As you can see that coefficients of x and y in the last two equations are same and the constants are 10 and 5. So, if we can make the coefficients of z also same which is k = 3, then these 2 equation will represent two separated parallel planes, which would never meet. Hence, no solution to the system of equations.

We don't need to spend time on doing Gaussian Elimination steps to figure this out. If you understand the geometry behind linear algebra, you can mark the correct answer in just 2 seconds.

Please focus on understanding the maths rather than just solving questions.

Comments

Yeah, this is good for some great observation, but procedure applies everywhere.

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I agree that procedure is everywhere, but a person who is applying the procedure need to realize what happens when a procedure gives some number as your answer.

Even if we apply Gaussian Elimination to get k = 3 for no solution, our knowledge will not grow if we don't realize that the 2 planes become parallel to each other in that case and that is why there is no solution.

Linear Algebra is best understood with the beautiful planar geometry running behind each and every results and theorems.

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Yes, but sometimes every one not able to visualize.

But Visualize the plane is very good and it improves the way of thinking.

I'm also not able to visualize so, much

thanks