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In mathematics, especially algebra, solving equations is a fundamental process. An equation states that two expressions are equal and often involves variables we need to determine. For example, in our problem, we have two equations based on the problem's conditions: the total gallons of gasoline sold and the total revenue from those sales. These equations can generally be written as:

1. The total gallons equation: \[ r + p = 1200 \]
2. The total revenue equation: \[ 3.85r + 4.05p = 4675 \]

To solve them, you must systematically work through substitution or elimination processes to isolate the variables and find their values. Always make sure each step logically follows the previous one to ensure the consistency and correctness of the solution.

Algebra word problems can initially seem daunting, but they're simply narratives describing situations where we apply algebra to find solutions. The key is to translate the words into mathematical expressions and equations.
In our problem, the key elements were:
- Regular gasoline costs \(3.85 per gallon
- Premium gasoline costs \)4.05 per gallon
- A combined total of 1200 gallons was sold
- Total revenue from the sales was $4675

By defining the variables and setting up equations, these scenarios are now easier to visualize and solve using algebra.

The substitution method is particularly effective for solving systems of linear equations. You solve one of the equations for one variable and then substitute that expression into the other equation. This simplifies the complex problem into a single-variable problem. Here's how it works for our problem:

First, solve the total gallons equation for \[ r \] :
\[ r = 1200 - p \]

Next, substitute this expression into the total revenue equation: \[ 3.85(1200 - p) + 4.05p = 4675 \]

By substituting, you reduce the problem to one equation with one variable that you can solve step-by-step. This systematic approach can be applied to many types of linear systems.

Elementary algebra forms the building blocks of more advanced mathematical concepts. It involves operations and principles helping us manipulate mathematical symbols to solve equations. Fundamental concepts include understanding variables, balancing equations, and applying arithmetic operations. In the original solution:

- Identifying variable relationships
- Setting up equations
- Applying operations like distribution \[ 3.85(1200 - p) \]
- Simplifying terms \[ 4620 - 3.85p + 4.05p \]
These are all basic yet crucial elements of elementary algebra. Grasping these foundational techniques makes solving more complicated mathematical problems straightforward.