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When approaching a problem involving a system of equations, the first step is to identify the variables. This process is called variables identification, and it's crucial because it helps define the unknowns we are trying to find. In this exercise, we need to determine the amount of money Keiko has in two different accounts.
We designate these unknowns by variables:

• Let \( x \) represent the amount of money in the smaller account.
• Let \( y \) represent the amount of money in the larger account.

By clearly defining the variables, it becomes easier to formulate equations that reflect the problem's conditions. This step is vital in correctly setting up your equation system, as it allows subsequent steps to be more straightforward and logical.

Once we have identified our variables, the next step is to express the scenario with a total sum equation. This equation reflects the total amount of money in both accounts combined, based on the information given.
Keiko has a total of \( \\(6500 \) invested in the two accounts. Thus, we know:

• The sum of the money in both accounts is \( \\)6500 \).
• This relationship is expressed mathematically as: \( x + y = 6500 \).

The total sum equation is essential because it ties the two accounts together under one constraint. By ensuring the equation represents the total correctly, we can check whether future calculations maintain a correct and consistent relationship between all elements involved.

To accurately describe all conditions in the problem, we need a difference equation. This type of equation captures how much more or less one variable is compared to another.
From the exercise, we're told that the larger account holds \( \\(800 \) more than the smaller one. This implies:

• The amount in the larger account exceeds the amount in the smaller account by \( \\)800 \).
• In equation form, this is represented as: \( y = x + 800 \).

The difference equation is key to understanding the exact relationship between the two account balances. It offers an additional constraint that works alongside the total sum equation to provide more depth, allowing us to solve for exact values of each account when combined with the total sum equation.

With the total sum and difference equations constructed, the next phase is equation modeling. This involves bringing together all these relationships into a cohesive system of equations.
The purpose is to solve for the variables that define our unknowns:

• We have the total sum equation: \( x + y = 6500 \).
• We also have the difference equation: \( y = x + 800 \).

These two equations together form a system of equations:\[\begin{align*}x + y &= 6500 \ y &= x + 800\end{align*}\]This system of equations provides a comprehensive model of the problem scenario. Although we are not required to solve it here, knowing how to model these equations prepares us to tackle simultaneous equation problems effectively. Equation modeling involves using logical analysis to convert real-world conditions into solvable mathematical representations.