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Understanding linear equations in one variable is crucial for solving real-world problems such as population projection. A linear equation in one variable is an equation that can be written in the standard form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable.

For instance, in the population projection problem, the equation for Greece's future population can be expressed as \(G(x) = 10,600,000 - 28,000x\). Here, the \( -28,000\) represents the annual population decrease, with \(x\) denoting the number of years passed. This is a linear equation with one variable, \(x\), which exhibits a constant rate of change—in this case, a decrease—in the population over time.

Mathematical modeling involves creating a mathematical representation of a real-life scenario to predict future outcomes or understand certain behaviors. In our exercise, the mathematical model is based on the populations of Greece and Belgium, which decrease linearly over time due to certain factors, symbolized here by the given projections.

The population model is represented by linear equations, which are derived from the given data points: \(G(x) = 10,600,000 - 28,000x\) for Greece and \(B(x) = 10,200,000 - 12,000x\) for Belgium. By creating these models, we convert a real demographic scenario into a mathematical problem that can be analyzed and solved using algebraic methods.

Solving equations is a foundational skill in algebra that involves finding the value(s) of the variable(s) that make the equation true. When we set the population equations of Greece and Belgium equal to each other, \(10,600,000 - 28,000x = 10,200,000 - 12,000x\), we're searching for the point where both countries have the same population.

The process usually involves isolating the variable on one side of the equation. For this problem, we combine like terms and simplify: \(16,000x = 400,000\), and then solve for \(x\) by dividing both sides by \(16,000\), which gives us \(x = 25\). This value of \(x\) represents the number of years after the year 2000 when the populations will match. By mastering the technique of solving linear equations, students can tackle a wide range of problems in various fields.