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# If a, b, and c are three different numbers and x/(b - c) = y/(c - a) =

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Math Expert
Joined: 02 Sep 2009
Posts: 58396
If a, b, and c are three different numbers and x/(b - c) = y/(c - a) =  [#permalink]

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07 Apr 2019, 21:33
00:00

Difficulty:

45% (medium)

Question Stats:

65% (02:02) correct 35% (02:17) wrong based on 23 sessions

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If a, b, and c are three different numbers and $$\frac{x}{b-c}=\frac{y}{c-a}=\frac{z}{a-b}$$, then what is the value of $$ax + by + cz$$ ?

(A) 0

(B) 1

(C) 2

(D) 3

(E) 4

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Intern
Joined: 09 Apr 2018
Posts: 32
Re: If a, b, and c are three different numbers and x/(b - c) = y/(c - a) =  [#permalink]

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07 Apr 2019, 22:14
In ax+by+cz, substituting y(b-c)/(c-a) for x and y(a-b)/(c-a) for z,
solving gives us 0/c-a

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Intern
Joined: 06 Apr 2019
Posts: 9
Re: If a, b, and c are three different numbers and x/(b - c) = y/(c - a) =  [#permalink]

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07 Apr 2019, 22:28
From given equation we know that a, b, c can't be equal so, Let a=1,b=2,c=3,
Then the given equation will become x=-y/2=z
Now put all the values of a, b, c in ax+by+cz
=x+2y+3z
Now as x=-y/2=z
The equation will become
=-y/2+2y-3y/2
=0

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Intern
Joined: 06 Apr 2019
Posts: 9
Re: If a, b, and c are three different numbers and x/(b - c) = y/(c - a) =  [#permalink]

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07 Apr 2019, 22:35
1
Another way solve it is put
X/(b-c) =Y/(c-a) =z/(a-b) =k
And put the values of x, y, z from above equation in ax+by+cz
It will become 0.
Hence A

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Joined: 18 Aug 2017
Posts: 5027
Location: India
Concentration: Sustainability, Marketing
GPA: 4
WE: Marketing (Energy and Utilities)
Re: If a, b, and c are three different numbers and x/(b - c) = y/(c - a) =  [#permalink]

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08 Apr 2019, 01:23
1
Bunuel wrote:
If a, b, and c are three different numbers and $$\frac{x}{b-c}=\frac{y}{c-a}=\frac{z}{a-b}$$, then what is the value of $$ax + by + cz$$ ?

(A) 0

(B) 1

(C) 2

(D) 3

(E) 4

let a=1,b=2 ,c = 3
and x=-1,y=2,z=-1
so $$ax + by + cz$$
IMO A ; 0
Senior Manager
Joined: 12 Sep 2017
Posts: 302
Re: If a, b, and c are three different numbers and x/(b - c) = y/(c - a) =  [#permalink]

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18 Apr 2019, 14:32
aankit

Hello!

Why are we sure that we can substitute the values with any number?

Kind regards!
Intern
Joined: 06 Apr 2019
Posts: 9
Re: If a, b, and c are three different numbers and x/(b - c) = y/(c - a) =  [#permalink]

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18 Apr 2019, 18:12
jfranciscocuencag wrote:
aankit

Hello!

Why are we sure that we can substitute the values with any number?

Kind regards!

as no. of equations is less than no. of variables and if the equation is correct with one or more solution then we can take any value of a,b,c provided they are not equal, these values have to satisfy the equation.
Re: If a, b, and c are three different numbers and x/(b - c) = y/(c - a) =   [#permalink] 18 Apr 2019, 18:12
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