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Intermediate algebra is a fundamental concept that helps us solve problems involving unknown values. In our train travel problem, we use variables to represent the speeds of the two trains. This allows us to form equations based on the given conditions. By using techniques like substitution or elimination, we can solve these equations to find the values of the unknown variables.

In this exercise, variables such as \(v_1\) and \(v_2\) are used for the speeds of the trains. Then, we create equations to represent the distances traveled by each train based on their speed and travel time. Intermediate algebra helps us break down the information and systematically solve the problem.

A system of equations is a set of two or more equations that have common variables. Solving systems of equations involves finding the values of the variables that satisfy all the equations simultaneously. This method is crucial in problems like train travel, where multiple conditions must be met at the same time.

In our case, we have two equations:
\[3v_1 + 2v_2 = 216\] \[3v_2 + 1.5v_1 = 216\]
These equations arise from considering two different travel scenarios where distances add up to the same value of 216 km. To solve this system, we can use methods such as the substitution method or the elimination method. Here, the elimination method was employed to eliminate one variable and solve for the other. Once one variable was found, it was substituted back into one of the original equations to find the other variable.

Distance and speed calculations are essential in understanding the relationship between how fast something is moving and how far it will travel over a given period. In problems involving moving objects like trains, it is common to use the formula: \[ \text{Distance} = \text{Speed} \times \text{Time} \]

For example, in our train travel problem:
- The first train travels for 3 hours at a speed of \(v_1\), covering a distance of \(d_1 = 3v_1\).
- The second train travels for 2 hours at a speed of \(v_2\), covering a distance of \(d_2 = 2v_2\).

When the two trains meet, the sum of the distances covered by both trains is equal to the total distance between the stations, which is 216 km. By setting up equations based on these relationships, we can solve for the speeds of the trains.

Understanding how to break down the problem and apply the distance formula is key to solving similar problems. This involves finding the time each train has traveled, multiplying it by their respective speeds, and ensuring the total distance matches the given problem condition.