We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
372
4
avatar+4286 

The equation \(y = \frac{x + A}{Bx + C}\), where \(A,B,\) and \(C\)  are integers, is shown below. What is \(A + B + C\)?

https://latex.artofproblemsolving.com/a/c/0/ac0cfb3f53a94da8604511d000667db714ff61b9.png

 Feb 5, 2018
 #1
avatar+8176 
+3

Here's one method...

 

There appears to be an asymptote at  x = 2 , so we know that when  x = 2 ,  Bx + C  =  0

 

B(2) + C  =  0

 

If we solve  \(y=\frac{x+A}{Bx+c}\)  for  x , we get  \(x=\frac{A-Cy}{By-1}\)

 

There appears to be an asymptote at  y = -1 , so we know that when  y = -1 ,  By - 1  =  0

 

B(-1) - 1  =  0

B  =  -1              Use this value for  B  to find  C .

 

(-1)(2) + C  =  0

C  =  2

 

And the graph passes through the point  (0, -2) , so...

 

\(-2 = \frac{0 + A}{-1(0) + 2} \\ -2=\frac{A}{2}\)

-4  =  A

 

Here's a graph of  \(y=\frac{x-4}{-1x+2}\)https://www.desmos.com/calculator/ibji5giqge

 

A + B + C   =   -4 + -1 + 2   =   -3

 Feb 5, 2018
 #2
avatar+101369 
+1

Very nice, hectictar.....!!!

 

 

cool cool cool

 Feb 5, 2018
 #3
avatar+4286 
+2

Amazing, hectictar!

 Feb 5, 2018
 #4
avatar+101369 
+3

Here's one more approach....

 

Note that the points    (0, - 2)  (3,1)  and (4,0)  are on the graph

 

So....we must have that

 

0 =  4 + A    ⇒  A  =  -4

 

And

 

-2  =  [0 + (-4)]  / [B (0) + C ]

 

-2  +  -4 / C

 

-2C  = - 4    ⇒  C  =  2

 

And

 

1  =  [ 3 + (-4) ] / [ B(3) + 2 ]

 

1 =  - 1 / [ 3B + 2]

 

3B + 2  =  - 1

 

3B =  -3  ⇒   B =  -1

 

So....A + B + C     =     -4 - 1 + 2    =    -3

 

 

cool cool cool

 Feb 6, 2018

11 Online Users

avatar
avatar