Question

Please solve using Cramer's rule
Give one value of t for which the system of equations
x +3y=ty
3x+y =tx
has infinitely many solutions.

Please provide me with a worked example ...kind regards.

I posted this problem a few days ago and got very helpful solutions using subtraction etc., Someone advised using cramers rule but i got stuck because the RHS of the equation is in the form tx and ty rather than numbers. Please help.

Answer 1

do you know how to find the determinant of a matrix?

Answer 2

in order to get infinite many solution you should get determinant =0

Answer 3

choose such a value of t for which the system determinants =0

Answer 4

simplifying
we can write as
x+3y-ty=0
3x-tx+y=0

Answer 5

@unseenoceans you getting the steps?

Answer 6

ok take t common in the above two equation to have something simple
x+y(3-t)=0
(3-t)x+y=0

Answer 7

\[D_y=\left|\begin{matrix}1&0\\3-t&0\end{matrix}\right|=0\]hm... this does seem strange to use cramers rule here

Answer 8

can you find determinant of above system ?

Answer 9

...but the above is always zero, so that would suggest that the value of t does not matter
now I'm confused :P

Answer 10

@TuringTest i think you missed t with y term in first equation.

Answer 11

in cramers rule you sub the RHS though

Answer 12

Yes , I know how to find determinant (ab-cd)

Answer 13

\[x+3y=ty,
3x+y=tx\]

Answer 14

this is from the problem sheet

Answer 15

\[\left[\begin{matrix}1 & 3-t \\ 3-t & 1\end{matrix}\right]\]

Answer 16

find the determinant of above ..let me know .you will get Quadratic equation
set the equation equal to zero find the two roots.
can you do this?

Answer 17

let me attempt

Answer 18

ok

Answer 19

is that 8-6t+t^2

Answer 20

roots are 2 and 4

Answer 21

I was advised when I first posted this problem that Cramers rule will be quickest. I am not so sure anymore

Answer 22

1-(3-t)(3-t)=1-9+6t-t^2=-8+6t-t^2=0
I don't think cramers rule is the fastest, it's easiest by inspection I would say

Answer 23

okay

Answer 24

use the above t value and you will have system of infinite solution.

Answer 25

Apart from the speed of solving the problem, can it be done with cramers rule, Turing?

Answer 26

you can also use Gauss jordon method to check..reduce the system to echelon form after putting t value and you will get one complete row of zeros

Answer 27

I am getting confused because sami's way works, but doesn't seem to be using cramers rule completely in the way I am used to

Answer 28

sami there are 2 t values : (2 and 4) the question asks for one value. Which one is the right one or are they both right?

Answer 29

it says "give one value" so I guess you could give either

Answer 30

cramer's rule is good for solving the system that have unique solution .for infinite use Gauss jordon.

Answer 31

I agree^

Answer 32

nice tip sami - do i reduce the second row to 0's?

Answer 33

How do you tell that above has infinite no solutions? DO you determine that from the question or is it because of the RHS : tx, ty?

Answer 34

the easiest way to check weather the system has infinite solutions is take determinent if its zero you will have infinite solutions lets use t=2
x+3y=2y
3x+y=2x
or
x+y=0
x+y=0
find determinant

Answer 35

also look at them these are both the equation of same line. and you may know that there are infinite no of solutions to a line.
we get solution when the lines intersects.
but
(parallel lines intersects at infinity)

Answer 36

yes, I understand now. Great explanation Sami....much appreciated ...I'll be looking out for you and Turing test next time! Kind regards

Answer 37

Once again thank you Sami-21 and Turing Test. Goodbye.

Answer 38

you are Most welcome:)
@unseenoceans .
i will be glad to help you.
that's what i do . i love to help students with their Questions .and make them Understand.