Algebra math?

Question

Find a and b, where x =not 1 and 2

((3x + 1)/((x - 1)(x - 2))) = (a/(x - 1)) + (b/(x - 2))

Answer 1

This is called partial fraction decomposition.
Multiply everything by (x-1)(x-2)
3x + 1 = a(x-2) + b(x-1)
Let x = 1
3(1) + 1 = a(1-2) + b(1-1)
3 + 1 = a(-1) + b(0)
4 = -a
-4 = a
Now let x = 2
3(2) + 1 = a(2-2) + b(2-1)
6 + 1 = a(0) + b(1)
7 = b
Answer: a = -4 and b = 7
Note: I used x = 1 and x = 2 because these required the shortest amount of work. You can use any two other values (like x = 0 and x = -1) and you would arrive at the same answer (it would just take more time).
Alternate way: try x = 0 and x = -1
Let x = 0, then
3(0) + 1 = a(0-2) + b(0-1)
0 + 1 = a(-2) + b(-1)
1 = -2a - b
Now let x = -1
3(-1) + 1 = a(-1-2) + b(-1-1)
-3 + 1 = a(-3) + b(-2)
-2 = -3a - 2b
You now have two equations:
1 = -2a - b
-2 = -3a - 2b
Multiply the first equation by -2
-2 = 4a + 2b
Now add the two equations together.
-2 = 4a + 2b
-2 = -3a - 2b
-4 = a
Replace a with -4 in either equation to find b.
1 = -2(-4) - b
1 = 8 - b
-7 = -b
7 = b
You can see that I have arrived at the same answer (but the second one required more work).

Answer 2

((3x + 1)/((x - 1)(x - 2))) = (a/(x - 1)) + (b/(x - 2))

3x + 1 = A(x - 2) + B(x - 1)

3(2) + 1 = A(2 - 2) + B(2 - 1)
6 + 1 = B
B = 7

3(1) + 1 = A(1 - 2) + B(1 - 1)
3 + 1 = -A
4 = -A
A = -4

ANS : a = -4 and b = 7

Complete Problem becomes
(-4/(x - 1)) + (7/(x - 2))

Answer 3

What's this? You have (x) (a) and (b). That's three variables!
You can't solve it with this little information.

I stopped at:
3x+1 = bx - b + ax - 2a. That's all i could come up with.

Edit:
Mathgirl, x can't be 1 nor 2. That's what i understood.

Answer 4

God will provide the answer if you ask him.