1/x+1/y=1 xy=6, then what is the value of x^2+y^2?

Question

Hi guys,

I am struggling with this one, can anyone help pls?

1/x + 1/y =1 xy=6, then what is the value of x^2 + y^2?

1. 12
2. 16
3. 18
4. 20
5. 24

Thanks!

Bunuel · Accepted Answer

\frac{1}{x}+\frac{1}{y}=1  -->  \frac{x+y}{xy}=1  -->  x+y=xy . Since  xy=6  then  x+y=6 .

Now,  x^2+y^2=(x+y)^2-2xy=6^2-2*6=24 .

Answer: E.

Splendidgirl666 · Answer

Got it! I forgot that a^2 + b^2 = (a+b)^2 - 2ab!

Thank you

PareshGmat · Answer

\frac{1}{x} + \frac{1}{y} = 1

x+y = xy

Squaring both sides

x^2 + 2(xy) + y^2 = (xy)^2

x^2 + y^2 = (xy)^2 - 2(xy)

Placing value of xy = 6

x^2 + y^2 = 36 - 12 = 24

Answer = 24

TimeTraveller · Answer

1/x + 1/y = 1; xy = 6.

1/x + 1/y = 1
x+y/xy = 1
x+y/6 = 1 (since xy = 6)
so, x+y = 6.

x^2+y^2 = (x+y)^2 - 2xy = (6)^2 - 2(6) = 36 - 12 = 24.

Ans (E).

iliavko · Answer

Hi everyone!

How come that x^2 + y^2 equals (x+y)^2 - 2xy ?

Is it like an algebraic identity formula? I haven't seen this one anywhere.

Thanks!

ENGRTOMBA2018 · Answer

It comes from rearranging terms in the formula:  (a+b)^2 = a^2+2ab+b^2  --->  a^2+b^2 = (a+b)^2-2ab

Hope this helps.

iliavko · Answer

Nice!!.. The stuff you discover at this forum

Thanks you so much! One more question, maybe it's silly, but.. What about difference of squares? a^2 - b^2 = (a+b)(a-b) what is the rationale behind it? Any formula that manipulated leads to it? Thank you!

Mathivanan Palraj · Answer

x*y=6
x+y=6
(x+y)^2=x^2+y^2+2*x*y
36=()+12
Therefore, the answer is 24

stonecold · Answer

See here => 1/x+1/y=1
so taking the lcm we get => x+y=xy
so (x+y)^2 =36
x^2+y^2=36-2xy=36-12 = 24
hence E

CLIMBTHELADDER · Answer

This is a cheeky little exercise.

The concepts that are tested: manipulating algebraic fractions, and square of a sum.

These are the steps I recommend to take, with the tested concepts mentioned in each step. 
 
1/x + 1/y = 1.  (we remember that with fractions with a different denominator, we first have to find the least common denominator)
  
y+x / xy =1

y+x = xy

y+x = 6

y^2+x^2  = 6^2

y^2+x^2  = 36.

(y+x)^2 = 36. (remember that  (y+x)^2 → x^2+2xy+y^2 
 
Now we can set up the equation with what we already have:

So x^2 + 2xy + y^2 = 36.  (Hence we also know that xy = 6. Let’s plug it in.

So x^2 + 2(6) + y^2 = 36 → x^2+12+y^2 = 36 → x^2 + y^2 = 36-12 = 24. Answer choice E.

ScottTargetTestPrep · Answer

We can multiply the first equation by xy and we have:

y + x = xy

Since xy = 6 we have:

y + x = 6

Squaring both sides we have:

x^2 + y^2+ 2xy = 36

x^2 + y^2 + 12 = 36

x^2 + y^2 = 24

Answer: E

hiranmay · Answer

1/x + 1/y =1 xy=6, then what is the value of x^2 + y^2?

1. 12
2. 16
3. 18
4. 20
5. 24 -->   correct : xy=6 & 1/x + 1/y =1 => x+y=xy=6=> (x+y)^2=6^2=>x^2+y^2+2xy=36=> x^2+y^2=24

preetamsaha · Answer

given, 1/x + 1/y = 1; xy = 6.

1/x + 1/y = 1
or,(x+y)/xy = 1
or,(x+y)/6 = 1 (since xy = 6)
so, x+y = 6.

now,x^2+y^2 = (x+y)^2 - 2xy = (6)^2 - 2(6) = 36 - 12 = 24.

correct answer E

bumpbot · Answer

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1/x+1/y=1 xy=6, then what is the value of x^2+y^2?

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1/x+1/y=1 xy=6, then what is the value of x^2+y^2?

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