IIT JEE , AIEEE Forum - Mathematics , Algebra

konichiwa2x

Solve the system of equations:

solve the system

This question can be done easily in many ways..But try to do it as few steps as possible.

vineetnegi

edited:

2x-y=1
replace it in 2 eq
can this solve it
the expression becomes very complicated

y^3+3+2y+ln(y^2+1+y)=0

if y solves ln(y^2+1+y)=0

and (y^3+3+2y)=0

then it solve the whole expression

for ln part

y^2+1+y=1

as ln(1)=0

y=0,-1

testing -1

we find that it solve the second exp

y=-1

x=0

.

konichiwa2x

well, it is progress.. But can you finish it?

computer001

y=-1,x=0

sandeepramesh

the method??? Dont tell me that was just a subsn

computer001

well 1st eqn gives 2x-y=1..in the 2nd eqn it seems lik a simple polynomial eqn.. v have ln(y^2 + y +1) if this returns some value it ob wont get cancelled wid other terms..i mean on keen obs tht wud b 2 unlikely to happen..so i put y^2 + y +1=1..hence the result :)

vineetnegi

i worked it on comp.
y=something like -.209574.....

computer001

y=-1,x=0

hash_include

is there any proper method to prove 2x-y - 1 anyway??

vineetnegi

you are right compu.
i mistyped the expression

konichiwa2x

clever computer001!

But can anyone provide a complete solution?

goldknight

From where did you get this question ?? mention some book not friend .

konichiwa2x

Solution:

Let

. then the first equation becomes

Clearly

is a solution.

We will show that there is no other solution: if

, Then

Therefore:

, which is impossible.

If

, then:

So

, which is again impossible.

Thus

is the only solution.

Hence

, i.e.

Put this in the second equation to get:

The easiest way to show that

is the only solution to this equation
is to show that

is strictly increasing in

that's quite easy:

(the last inequality is true because we know that if

and

, then

)

So we have

and

thus

is the only solution to the system.

Quote from goldknight:

From where did you get this question ?? mention some book not friend .

From quite an old book. "Higher Mathematics" by A.G.Wells.

computer001

@ KONI:
but how do u show tht y=-1 is a soln

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