SOLUTION: {{{(a(x+1)+b(x-1))/(x-2)=2+(1/(x-2))}}} The equation above is true for all values of x≠2, where a and b are constats. What is the value of a? A) -1/2 B) 2 C) 3 D) 4

Algebra ->  Expressions -> SOLUTION: {{{(a(x+1)+b(x-1))/(x-2)=2+(1/(x-2))}}} The equation above is true for all values of x≠2, where a and b are constats. What is the value of a? A) -1/2 B) 2 C) 3 D) 4      Log On


   

Question 1043622: %28a%28x%2B1%29%2Bb%28x-1%29%29%2F%28x-2%29=2%2B%281%2F%28x-2%29%29
The equation above is true for all values of x≠2, where a and b are constats. What is the value of a?
A) -1/2
B) 2
C) 3
D) 4
Found 2 solutions by jim_thompson5910, rothauserc:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!



For now, I'm going to simplify the right side only


%28a%28x%2B1%29%2Bb%28x-1%29%29%2F%28x-2%29=2%2B%281%2F%28x-2%29%29








%28a%28x%2B1%29%2Bb%28x-1%29%29%2F%28x-2%29=%282x-4%29%2F%28x-2%29%2B%281%2F%28x-2%29%29


%28a%28x%2B1%29%2Bb%28x-1%29%29%2F%28x-2%29=%282x-4%2B1%29%2F%28x-2%29%29


%28a%28x%2B1%29%2Bb%28x-1%29%29%2F%28x-2%29=%282x-3%29%2F%28x-2%29%29


Now I'm going to rearrange terms on the left side


%28a%28x%2B1%29%2Bb%28x-1%29%29%2F%28x-2%29=%282x-3%29%2F%28x-2%29%29


%28ax%2Ba%2Bbx-b%29%2F%28x-2%29=%282x-3%29%2F%28x-2%29%29


%28ax%2Bbx%2Ba-b%29%2F%28x-2%29=%282x-3%29%2F%28x-2%29%29


%28%28ax%2Bbx%29%2B%28a-b%29%29%2F%28x-2%29=%282x-3%29%2F%28x-2%29%29


%28%28a%2Bb%29x%2B%28a-b%29%29%2F%28x-2%29=%282x-3%29%2F%28x-2%29%29


Take note that the x term on the left side numerator is %28a%2Bb%29x. The x term on the right side numerator is 2x. Equate the coefficients to get this equation a%2Bb=2


The constant terms on the left side numerator is a-b. The constant terms on the right side numerator is -3. Equate the terms to get the equation a-b=-3


--------------------------------------------------------------


We have this system of equations


system%28a%2Bb=2%2Ca-b=-3%29


Let's add the two equations up


a+b=2
a-b=-3
-------
2a+0b=-1


So 2a=-1. Divide both sides by 2 to isolate 'a'. Doing this will lead to a+=+-1%2F2


So the answer is choice A) -1/2



Answer by rothauserc(4718) About Me  (Show Source):
You can put this solution on YOUR website!
2 + ( 1 / (x-2) ) = (2x-3) / (x-2), then
:
a(x+1) + b(x-1) = 2x-3
:
ax+a +bx-b = 2x-3
:
we have two equations in 2 unknowns
:
1) ax +bx = 2x
2) a -b = -3
:
solve 2) for b
:
b = a +3
:
substitute for b in equation 1)
:
ax +(a+3)x = 2x
:
2ax = -x
:
************
a = -0.5
answer is A)
************
:
check our answer
a = -0.5, then b = -0.5 +3 = 2.5
:
substitute for a and b in the original equation
:
(-0.5(x+1) + 2.5(x-1)) / (x-2) =
:
(-0.5x -0.5 +2.5x -2.5) / (x-2) =
:
(2x - 3) / (x-2) = 2 + ( 1 / (x-2) ) = (2x -3) / (x-2)
:
our answer checks :-)
: