To solve this problem step-by-step, we have:
1. Examine the initial equation:
[tex]\[
\frac{3 x}{2 x-6}+\frac{9}{6-2 x}
\][/tex]
2. Notice that [tex]\(6 - 2x = -(2x - 6)\)[/tex], so we can rewrite the second fraction:
[tex]\[
\frac{3 x}{2 x-6}+\frac{9}{a(2 x-6)}
\][/tex]
3. Identifying [tex]\(a\)[/tex]:
Since [tex]\((6 - 2x) = -(2x - 6)\)[/tex], it follows that [tex]\(a = -1\)[/tex]:
[tex]\[
a = -1
\][/tex]
4. Updating the expression, we have:
[tex]\[
\frac{3 x}{2 x-6}+\frac{9}{-1(2 x-6)}
\][/tex]
5. Simplify the second fraction by multiplying the numerator and denominator by -1:
[tex]\[
\frac{3 x}{2 x-6}+\frac{9}{-(2 x-6)} = \frac{3 x}{2 x-6} + \frac{9}{-1(2 x-6)} = \frac{3 x}{2 x-6} + \frac{-9}{2 x-6}
\][/tex]
6. Combining the fractions:
[tex]\[
\frac{3 x}{2 x-6}+\frac{-9}{2 x-6} = \frac{3 x - 9}{2 x-6}
\][/tex]
7. Identifying [tex]\(b\)[/tex] and [tex]\(c\)[/tex]:
Since [tex]\(3x - 9\)[/tex] can be rewritten as [tex]\(3(x - 3)\)[/tex]:
[tex]\[
b = 9 \quad \text{and} \quad c = 3
\][/tex]
8. Further simplifying the numerator:
[tex]\[
\frac{3(x - 3)}{2 x-6}
\][/tex]
9. Factor [tex]\(d\)[/tex], [tex]\(e\)[/tex], and [tex]\(f\)[/tex]:
[tex]\[
d = -3 \quad e = -1 \quad f = -2
\][/tex]
10. Lastly, simplify the fractions:
[tex]\[
\frac{-3(x - -1)}{-2(x-3)} = 1
\][/tex]
11. Identifying [tex]\(g\)[/tex]:
[tex]\[
g = 1.0
\][/tex]
Therefore, the values are:
[tex]\[
a = -1, \quad b = 9, \quad c = 3, \quad d = -3, \quad e = -1, \quad f = -2, \quad g = 1.0
\][/tex]
