2(4x+3) a b
----------- ≡ (identically equal to) ------ + ------
(x-3)(x+7) x-3 x+7
(2(4x+3))/((x-3)(x+7)) ≡ a/(x-3) + b/(x+7)
Sorry, the spaces messed up, but the question/equation is on the bottom.
Looks like you are studying "partial fractions"
(2(4x+3))/((x-3)(x+7)) ≡ a/(x-3) + b/(x+7)
(8x + 6)/((x-3)(x+7)) ≡ (a(x+7) + b(x-3) )/(((x-3)(x+7))
multiply by (x-3)(x+7)
8x + 6 ≡ a(x+7) + b(x-3)
this must be true for all values of x
let x = 3, ---> 24+6 = 10a + 0
a = 3
let x = -7, ---> -56+6 = 0 - 10b
b = 5
you might have learned to expand 8x + 6 ≡ a(x+7) + b(x-3) and then comparing like terms,
8x + 6 = ax + 7a + bx - 3b
8x + 6 = x(a+b) + 7a-3b
then a + b = 8 ---> a = -b+8
and 7a - 3b = 6
by substitution:
7(-b+8) - 3b = 6
-7b + 56 - 3b = 6
-10b = -50
b = 5 , then a = -5+8 = 3
I find the method I used much simpler.
Thank you very much.
To prove that the given expression is identically equal to the equation (a/(x-3)) + (b/(x+7)), we need to first find the values of a and b that satisfy the equation.
To find the values of a and b, we can start by multiplying both sides of the equation by the denominator of the expression on the left side, which is (x-3)(x+7).
This would give us:
2(4x+3) = a(x+7) + b(x-3)
Expanding the right side:
8x + 6 = ax + 7a + bx - 3b
Now, let's simplify the expression by collecting like terms.
First, let's collect the terms which have x:
8x = ax + bx
Rearranging this equation:
8x = (a + b)x
Since both sides of the equation are equal, the coefficients of x must be equal as well. Therefore, we can conclude that:
a + b = 8 ----(1)
Next, let's collect the constant terms:
6 = 7a - 3b
Rearranging this equation:
7a - 3b = 6
Now, we have two equations that we can solve to find the values of a and b.
First, let's rearrange equation (1):
b = 8 - a
Substituting this value of b into the equation (2):
7a - 3(8 - a) = 6
Expanding and simplifying:
7a - 24 + 3a = 6
Combining like terms:
10a - 24 = 6
Adding 24 to both sides:
10a = 30
Dividing by 10:
a = 3
Now, substitute the value of a back into equation (1):
b = 8 - 3
b = 5
Therefore, the values of a and b are a = 3 and b = 5.
So, (2(4x+3))/((x-3)(x+7)) is identically equal to (3/(x-3)) + (5/(x+7)).