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If (3x + 2y - 22)^2 + (4x - 5y + 9)^2 = 0 and 5x - 4y = 0, then what

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Bunuel
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If (3x + 2y - 22)^2 + (4x - 5y + 9)^2 = 0 and 5x - 4y = 0, then what [#permalink] New post   29 May 2020, 10:07
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If \((3x + 2y - 22)^2 + (4x - 5y + 9)^2 = 0\) and \(5x - 4y = 0\), then what is the value of x + y ?

A. 7
B. 8
C. 9
D. 11
E. 13
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C
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BrentGMATPrepNow
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If (3x + 2y - 22)^2 + (4x - 5y + 9)^2 = 0 and 5x - 4y = 0, then what [#permalink] New post   29 May 2020, 10:45
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Bunuel wrote:
If \((3x + 2y - 22)^2 + (4x - 5y + 9)^2 = 0\) and \(5x - 4y = 0\), then what is the value of x + y ?

A. 7
B. 8
C. 9
D. 11
E. 13


Key concept: k² ≥ 0, for all values of k.

So, if something² + (something else)² = 0, then we can be certain that: something = 0 and (something else) = 0

Likewise, since (3x + 2y - 22)² + (4x - 5y + 9)² = 0 , we know that 3x + 2y - 22 = 0 and 4x - 5y + 9 = 0

I'm going to ignore the first equation, 3x + 2y - 22 = 0

Instead, notice that, if we take the other equation, 4x - 5y + 9 = 0 and combine it with the given equation, 5x - 4y = 0, we can quickly find the value of x + y without having to solve for the individual values of x and y.

We have:
5x - 4y = 0
4x - 5y + 9 = 0

Rewrite the bottom equation as follows:
5x - 4y = 0
4x - 5y = -9

Subtract the bottom equation from the top equation to get: x + y = 9

Answer: C

Cheers,
Brent
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Lipun
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Re: If (3x + 2y - 22)^2 + (4x - 5y + 9)^2 = 0 and 5x - 4y = 0, then what [#permalink] New post   29 May 2020, 10:18
Sum of two squares is zero. \(x^2+y^2=0\) => x,y=0.
Solving for \(3x+2y-22=0\) and \(4x-5y+9=0\) we obtain, x=4, y=5.

Sum=9
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Re: If (3x + 2y - 22)^2 + (4x - 5y + 9)^2 = 0 and 5x - 4y = 0, then what [#permalink] New post   29 May 2020, 10:52
Bunuel wrote:
If \((3x + 2y - 22)^2 + (4x - 5y + 9)^2 = 0\) and \(5x - 4y = 0\), then what is the value of x + y ?

A. 7
B. 8
C. 9
D. 11
E. 13


Asked: If \((3x + 2y - 22)^2 + (4x - 5y + 9)^2 = 0\) and \(5x - 4y = 0\), then what is the value of x + y ?

\((3x + 2y - 22)^2 + (4x - 5y + 9)^2 = 0\)
\((3x + 2y - 22)^2 = - (4x - 5y + 9)^2 = 0\) ; Since \(x^2>=0\)
3x + 2y - 22 = 0 (1)
4x - 5y + 9 = 0 (2)
5x - 4y = 0 (3)

(2) + (3)
9x - 9y + 9 = 0
x - y + 1 = 0
4y/5 - y + 1 = 0
y = 5
x = 4
x + y = 9

IMO C
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Re: If (3x + 2y - 22)^2 + (4x - 5y + 9)^2 = 0 and 5x - 4y = 0, then what [#permalink] New post   30 May 2020, 04:10
Given a^2 + b^2 =0
==> a=0 & b=0
==> 3x+32-22 = 0. ---1
& 4x -5y+9 = 0. ---2
Given: 5x -4y = 0 ----3

The question does not ask us to find individual values of x & y but just the sum of x & y

Now eqn 3 - eqn 2
==>x + y -9 = 0
or x +y = 9

C
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nsdl1985
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Re: If (3x + 2y - 22)^2 + (4x - 5y + 9)^2 = 0 and 5x - 4y = 0, then what [#permalink] New post   03 Jun 2021, 23:29
Bunuel wrote:
If \((3x + 2y - 22)^2 + (4x - 5y + 9)^2 = 0\) and \(5x - 4y = 0\), then what is the value of x + y ?

A. 7
B. 8
C. 9
D. 11
E. 13


Looking at answer choices and also looking at 5x-4y=0, we know that:-

x=4 and y=5 since 5x-4y=0

also had x=-4 and y=5 which also leads to 5x-4y=0, but then x+y=1 which is not the answer choice.
Thus x=4 and y=5. Therefore x+y=5.

Now we know what is the answer, do we need to plug in values of x=4 & y=5 in the equation (3x + 2y - 22)^2 + (4x - 5y + 9)^2 = 0?

Or is there something I am missing?
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Re: If (3x + 2y - 22)^2 + (4x - 5y + 9)^2 = 0 and 5x - 4y = 0, then what [#permalink] New post   20 Jan 2024, 04:15
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Re: If (3x + 2y - 22)^2 + (4x - 5y + 9)^2 = 0 and 5x - 4y = 0, then what [#permalink]
20 Jan 2024, 04:15
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