Question

enter the values for the highlighted variables to complete the steps to find the sum:
\\(\frac{3x}{2x - 6}+\frac{9}{6 - 2x}=\frac{3x}{2x - 6}+\frac{9}{a(2x - 6)}\\)
\\(=\frac{3x}{2x - 6}+\frac{b}{2x - 6}\\)
\\(=\frac{3x - c}{2x - 6}\\)
\\(=\frac{d(x - e)}{f(x - 3)}\\)
\\(= g\\)
\\(a = \boxed{-1}\\)
\\(b = \boxed{-9}\\)
\\(c = \boxed{9}\\)
\\(d = \boxed{3}\\)
\\(e = \boxed{}\\)
\\(f = \boxed{}\\)
\\(g = \boxed{}\\)

Explanation:

Step1: Identify numerator for subtraction

From $\frac{3x}{2x-6} + \frac{-9}{2x-6} = \frac{3x - c}{2x-6}$, match numerators:
$3x - 9 = 3x - c$ so $c=9$ (given). Now simplify numerator $3x - 9$.

Step2: Factor numerator

Factor out $d=3$ from $3x-9$:
$3x - 9 = 3(x - 3)$
Compare to $d(x-e)=3(x-e)$, so $e=3$.

Step3: Factor denominator

Factor $2x-6$:
$2x-6 = 2(x-3)$
Compare to $f(x-3)$, so $f=2$.

Step4: Simplify the fraction

Cancel $(x-3)$ (where $x
eq3$):
$\frac{3(x-3)}{2(x-3)} = \frac{3}{2}$
So $g=\frac{3}{2}$.

Answer:

$e=3$, $f=2$, $g=\frac{3}{2}$