Determining Rachel's Average: Solving Weighted Grade Percentages

First, identify the percentages associated with each grade, with the knowledge that all weights should total 100%.

We have:

• First exam: $95$ with $20\%$

• Second exam: $89$ with $15\%$

• Third exam: $78$ with $X\%$

• Fourth exam: $92$ with the remaining percentage.

Since the weights must sum to $100\%$, the equation becomes:
$20 + 15 + X + \text{remaining} = 100$.
This gives us:

$\text{remaining} = 100 - (20 + 15 + X)$

$\text{remaining} = 65 - X$

Now find the weighted average using:

$\left( 95 \times 0.20 \right) + \left( 89 \times 0.15 \right) + \left( 78 \times \frac{X}{100} \right) + \left( 92 \times \frac{(65-X)}{100} \right)$

Simplifying each term, we have:

$\begin{aligned} 95 \times 0.20 & = 19,\\ 89 \times 0.15 & = 13.35,\\ 78 \times \frac{X}{100} & = 0.78X,\\ 92 \times \frac{65-X}{100} & = 92 \times (0.65 - \frac{X}{100}) = 59.8 - 0.92X. \end{aligned}$

Adding these components yields:

$19 + 13.35 + 0.78X + 59.8 - 0.92X$.

Combine like terms to simplify further:

$92.15 - 0.14X$.

Therefore, Rachel's average grade can be expressed as $92.15 - 0.14X$.