If a/(b+c) = b/(c+a) = c/(a+b) = r,then r cannot any value except?

Question

If a/(b+c) = b/(c+a) = c/(a+b) = r,then r cannot any value except
(a) 1/2
(b) -1
(c) 1/2 or -1
(d) -1/2 or -1
The answer is c but how

Answer 1

a/(b+c)=b/(c+a) => a(c+a)=b(c+a)
ac+a^2=bc+b^2
a^2-b^2=-ac+bc
(a-b)(a+b)=-c(a-b)
1) a-b=0
a=b
or 2) (a-b)(a+b)=-c(a-b) /:(a-b)
=> a+b=-c
a+b+c=0
=> a/(b+c)=a/(-a)=r=-1

In the same way with b/(c+a)=c/(a+b) =>b=c or a+b+c=0
from b=c and a=c =>a/(b+c)=a/2a=1/2
that's all! if you have some questions mail me...

Answer 2

Equate the last 2 parts of the statement by the 1st statement to make 2 equations
a/(b + c) = b/(c + a)
a/(b + c) = c/(a + b)

Multiply each by their LCD
a² + ac = b² + bc
a² + ab = c² + bc

Subtract the 1st equation from the 2nd equation
c² - b² = ab - ac

Transpose to the left
c² - b² + ac - ab = 0

Factor monomially and by special factoring
(c + b)(c - b) + a(c - b) = 0

Factor the groups
(c - b)(c + b + a) = 0

There are 2 cases
c - b = 0 or c + b + a = 0
Thus,
b = c (Let it be case A) or a + b + c = 0 (Let it be case B)


---case A---
Since b = c and
a² + ac = b² + bc,

Substitute b for all c
a² + ab = b² + b²

Add and transpose to the left
a² + ab - 2b² = 0

factor
(a - b)(a + 2b) = 0

There are 2 cases for case A
a = b or a = -2b

For a = b = c, The value for r is r = a/(b + c) = a/(a + a) = a/2a = 1/2
For b = c,a = -2b, The value for r is r = a/(b + c) = (-2b)/(b + b) = -2b/2b = -1


---case B---
a + b + c = 0
Thus,
b + c = -a
The value for r is r = a/(b + c) = a/(-a) = -1

Thus, the only values for r that we got is 1/2 (from case A) and -1 (from both cases). Therefore, the answer is (c) 1/2 or -1.

(Note that there are some undefined values for r, specifically when b + c = 0, a + c = 0, and/or a + b = 0, but this does not affect our answer.)


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Answer 3

Ok, all these answers would burst a Math novice's head blood vessels, so I come with a simple logic.
The answer is C, as everyone knows, and until u do a whole lot of manipulations, u might not know how they come up with the answer. But you can easily see that the answer is correct.
The C options show 1/2 and -1. Now, 1/2 could be written as 1/ (1+1), and 1/2 could be re-written as 1/ (1/2 + 1/2). In each instance, let each variable have one of the numbers, and try plugging them in. It'll work for both options. now, if u decide to try any other combo of numbers, it won't work for all of the variables.
Try it, it works!

Answer 4

To explain r=1/2 is direct, when a=b=c, then sum of any 2 is the twice of any of them.
To explain r=-1, need to use complex numbers.
a=b=c=e^(ix), x=0, 2pi/3, 4pi/3
Visualize these 3 points in the complex plane, you will see that the addition of any two at the 180 degree rotation of the remaining one, and this rotation corresponds to a muliply by -1.
thus a = -1 x (b+c), b = -1 x (c+a) and c = -1 x (a+b).

Answer 5

a/(b+c) = b/(c+a) -> a^2 + ac = b^2 + bc (1)
b/(c+a) = c/(a+b) -> c^2 + ac = b^2 + ab (2)

(1) - (2) -> a^2 - c^2 = bc - ab
-> (a-c)*(a+c) = b*(c-a)
-> a = c (both sides are 0) or a + c = -b (cancel the random factor a-c)

If a = c then similarly we have b = c -> r = a/(b+c) = c/(c+c) = 1/2
If a+c = -b -> r = b/(c+a) = b/-b = -1

Answer 6

in 607 b.c the northern kingdom became overthrown through the assyrians in 587 the southern kingdom became overthrown through the Babylonians bear in mind the Hebrew year is 360 days. we've 365 and 1 / 4 days in our year, do not beat your self up over a year the following and there which will be out of kilter

Answer 7

I a/(b+c)=b/(a+c) => a2+ac=b2+bc
II a/(b+c)=c/(a+b) => a2+ab=c2+bc
III b/(a+c)=c/(a+b) => b2+ab=c2+ac

I,II => a(b-c)=c2-b2 => a(b-c)=(-b-c)(b-c)=> b=c or a+b+c=0
I, III=>b(c-a)=a2-c2 => b(c-a)=(-a-c)(c-a)=> c=a or a+b+c=0
II,III=>c(b-a)=a2-b2 => c(b-a)=(-b-a)(b-a)=> b=a or a+b+c=0

so you have to way
1- a=b=c => r=a/(b+c)=a/(a+a)=1/2=0.5
2- a+b+c=0 => r =a/(b+c)=a/(-a)=-1

Answer 8

R If Then

Answer 9

yes
the answer is C