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\\)
\\(n = \square\\)
\\(b = \square\\)
\\(c = \square\\)
\\(d = \square\\)
\\(e = \square\\)
\\(f = \square\\)
\\(g = \square\\)
done

Explanation:

Step1: Rewrite denominator to match

Notice that $6-2x = -(2x-6)$, so $a=-1$.

Step2: Adjust numerator for common denominator

Substitute $a=-1$: $\frac{9}{-1(2x-6)}=\frac{-9}{2x-6}$, so $b=-9$.

Step3: Combine numerators

$\frac{3x}{2x-6}+\frac{-9}{2x-6}=\frac{3x-9}{2x-6}$, so $c=9$.

Step4: Factor numerator

Factor $3$ from $3x-9$: $3x-9=3(x-3)$, so $d=3$.

Step5: Factor denominator

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

Step6: Simplify the fraction

Cancel $(x-3)$: $\frac{3(x-3)}{2(x-3)}=\frac{3}{2}$, so $g=\frac{3}{2}$.

Answer:

$a=-1$
$b=-9$
$c=9$
$d=3$
$f=2$
$g=\frac{3}{2}$