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To determine how many hours someone works in a year, we start by considering the number of hours worked each week and subtracting the weeks taken off for vacation. For example, if a person works a certain number of hours per week, say 40, and they take two weeks off for vacation, you would multiply the weekly hours by the weeks worked in a year, which is generally 52 weeks minus the vacation weeks.

If Max works for \(x\) hours per week and takes a 3-week vacation, he doesn't work during those vacation weeks. Thus, he works for \(52 - 3 = 49\) weeks a year.

Max's total work hours for the year can be calculated using the expression \(49 \times x\). This effectively accounts for the vacation time while providing a clear picture of annual work contributions.

Mindy's work calculation would follow a similar process, considering her specific work hours per week and vacation weeks. The key here is understanding that subtracting vacation time from the full year is crucial when calculating accurate annual work hours.

Vacation time subtraction is an essential part of calculating total work hours over a year. When a person takes a vacation, those weeks do not contribute to work hours, affecting the total calculation. Hence, accounting for vacation accurately helps in finding out real work contributions annually.

For Max, with a 3-week vacation, you subtract these weeks from the total 52 weeks in a year to determine the actual working weeks. This leaves Max with 49 working weeks, hence the equation \(52 - 3 = 49\).

This method of subtraction ensures that we do not overcount work hours by including time when the person is not actually working. It's a straightforward operation but crucial for getting precise results.

Similarly, Mindy has a 4-week vacation. Therefore, her working weeks are calculated as \(52 - 4 = 48\). This equation becomes part of the larger algebraic expression used to determine their total annual work hours.

Algebraic expressions are mathematical phrases that can include numbers, variables, and operators. They are vital in word problems for representing complex relationships simply and clearly.In this exercise, the algebraic expressions represent the total work hours for Max and Mindy after accounting for vacations. For Max, the expression \(49 \times x\) corresponds to his yearly work hours, with \(x\) representing his weekly hours.

Similarly, Mindy's yearly work hours can be expressed as \(48 \times y\), with \(y\) representing her weekly work commitment. Each expression effectively condenses the calculation of annual work hours into a simple formula.The entire problem combines these individual expressions into \(49x + 48y\), which represents their total combined annual work hours. Using these expressions allows us to handle variables and constants efficiently, solving for unknowns, and understanding how each component contributes to the whole.