Enter the values for the highlighted variables to complete the steps to find the sum: \( \begin{aligned} \frac{3 x}{2 x-6}+\frac{9}{6-2 x} & =\frac{3 x}{2 x-6}+\frac{9}{a(2 x-6)} \\ & =\frac{3 x}{2 x-6}+\frac{3 x-6}{2 x-6} \\ & =\frac{3 x-6}{2 x-6} \\ & =\frac{d(x-e)}{f(x-3)} \\ & =g\end{aligned} \)

To find the values for the highlighted variables in the given expression, let's analyze the steps one by one. 1. **Identify the common denominator**: The first step shows that we have two fractions: \[ \frac{3x}{2x-6} + \frac{9}{6-2x} \] Notice that \(6 - 2x\) can be rewritten as \(-(2x - 6)\) or \(-1(2x - 6)\). Thus, we can express the second fraction as: \[ \frac{9}{6 - 2x} = \frac{9}{-1(2x - 6)} = -\frac{9}{2x - 6} \] Therefore, the common denominator is \(2x - 6\). 2. **Rewrite the second fraction**: The second fraction can be rewritten as: \[ \frac{9}{6 - 2x} = \frac{9}{-1(2x - 6)} = -\frac{9}{2x - 6} \] This means we can express the sum as: \[ \frac{3x}{2x - 6} - \frac{9}{2x - 6} = \frac{3x - 9}{2x - 6} \] 3. **Simplify the numerator**: The numerator \(3x - 9\) can be factored: \[ 3(x - 3) \] Thus, we have: \[ \frac{3(x - 3)}{2x - 6} \] 4. **Factor the denominator**: The denominator \(2x - 6\) can also be factored: \[ 2(x - 3) \] Therefore, we can rewrite the expression as: \[ \frac{3(x - 3)}{2(x - 3)} \] 5. **Cancel common factors**: The \(x - 3\) terms cancel out (assuming \(x \neq 3\)): \[ \frac{3}{2} \] Now, let's assign the values to the highlighted variables: - For \(a\), since we expressed the second fraction as \(\frac{9}{a(2x - 6)}\), we can see that \(a = -1\). - For \(d\), since we have \(\frac{d(x - e)}{f(x - 3)}\) and we simplified to \(\frac{3}{2}\), we can set \(d = 3\) and \(f = 2\). - For \(e\), since we factored out \(x - 3\), we have \(e = 3\). - Finally, \(g\) is the simplified result, which is \(\frac{3}{2}\). Thus, the values are: - \(a = -1\) - \(d = 3\) - \(e = 3\) - \(f = 2\) - \(g = \frac{3}{2}\)