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# If x + y = 7 and x^2 + y^2 = 25, then which one of the following equal

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Joined: 02 Sep 2009
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If x + y = 7 and x^2 + y^2 = 25, then which one of the following equal  [#permalink]

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07 Apr 2019, 21:30
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15% (low)

Question Stats:

88% (01:17) correct 12% (02:15) wrong based on 26 sessions

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If $$x + y = 7$$ and $$x^2 + y^2 = 25$$, then which one of the following equals the value of $$x^3 + y^3$$ ?

(A) 7

(B) 25

(C) 35

(D) 65

(E) 91

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Re: If x + y = 7 and x^2 + y^2 = 25, then which one of the following equal  [#permalink]

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08 Apr 2019, 01:16
Bunuel wrote:
If $$x + y = 7$$ and $$x^2 + y^2 = 25$$, then which one of the following equals the value of $$x^3 + y^3$$ ?

(A) 7

(B) 25

(C) 35

(D) 65

(E) 91

$$x^2 + y^2 = 25$$
solve with x+y=7
we get y=4,3 and x= 3,4
x^3+y^3 = 27+64 ; 91
IMO E
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Re: If x + y = 7 and x^2 + y^2 = 25, then which one of the following equal  [#permalink]

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08 Apr 2019, 01:41
the smart method is stated above by archit.
However, solving algebraically:
(x+y)²=x²+y²+2xy
xy=12.

(x+y)³=x³+y³+3xy(x+y)
substitute the values to get
7³=x³+y³+3*12*7
x³+y³=91

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Re: If x + y = 7 and x^2 + y^2 = 25, then which one of the following equal  [#permalink]

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08 Apr 2019, 07:34
X^3 + y^3 = (X+y)(x^2+y^2-xy) --(1)

(X+y)^2 = x^2+y^2+2xy
49-25=2xy
2xy = 24
Xy=12

Substitute xy=12.
7(25-12)
7*13 = 91..

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Re: If x + y = 7 and x^2 + y^2 = 25, then which one of the following equal  [#permalink]

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08 Apr 2019, 07:44
$$x + y = 7$$
$$x^2 + y^2 = 25$$

$$7$$ as sum of two numbers is given by: 4+3; 6+1; 5+2 ...

If the second equation is verified, pick them; otherwise, reject them.

$$4^2 + 3^2 = 16$$? Yes. Pick them.

$$4^3 + 3^3 = 64 + 27 = 91$$. Correct answer: E.
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Re: If x + y = 7 and x^2 + y^2 = 25, then which one of the following equal  [#permalink]

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08 Apr 2019, 09:24
Top Contributor
Bunuel wrote:
If $$x + y = 7$$ and $$x^2 + y^2 = 25$$, then which one of the following equals the value of $$x^3 + y^3$$ ?

(A) 7

(B) 25

(C) 35

(D) 65

(E) 91

----ASIDE---------------
We COULD use some algebra to solve the question.
First rewrite the first equation as x = 7 - y
Then take second equation and replace x with 7 - y to get: (7 - y)² + y² = 25
When we eventually solve the system, we get two possible solutions:
Solution #1: x = 3 and y = 4
Solution #2: x = 4 and y = 3
-----------------------------

Solving the question is the above way takes time. So, before resorting to algebra, we might first see if we can solve the question through inspection.
If we do so, we might quickly realize that one value must be 3 and the other must be 4.

So, x³ + y³ = 3³ + 4³
= 27 + 64
= 91

Cheers,
Brent
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Re: If x + y = 7 and x^2 + y^2 = 25, then which one of the following equal  [#permalink]

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10 Apr 2019, 17:48
Bunuel wrote:
If $$x + y = 7$$ and $$x^2 + y^2 = 25$$, then which one of the following equals the value of $$x^3 + y^3$$ ?

(A) 7

(B) 25

(C) 35

(D) 65

(E) 91

Since x + y = 7, (x + y)^2 = 7^2 → x^2 + 2xy + y^2 = 49. Since x^2 + y^2 = 25, we have 25 + 2xy = 49 → 2xy = 24 → xy = 12.

Recall that x^3 + y^3 factored as (x + y)(x^2 - xy + y^2), so we have:

x^3 + y^3 = (7)(25 - 12) = 7(13) = 91

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Re: If x + y = 7 and x^2 + y^2 = 25, then which one of the following equal   [#permalink] 10 Apr 2019, 17:48
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