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An objective function is a mathematical expression used to represent the goal of a specific problem. Here, our goal is to determine the total number of calories burned. We introduce two variables: running for a certain number of minutes (\(x\)) and lifting weights for a number of minutes (\(y\)).
We know the caloric burn rates: 10 calories per minute for running and 8 calories per minute for lifting weights.
By summing up the contributions of both activities, we can create a function representing the total calories burned. This function is given by:
\[ z = f(x, y) = 10x + 8y \] This equation lets us calculate the total calories burned given any potential values for minutes spent running and lifting weights, thus serving as the objective function for the athlete’s workout.

Calories burned refers to the amount of energy expended through physical activities. For this athlete, counting the calories from running and lifting weights is crucial:

• Running burns 10 calories per minute.
• Lifting weights burns 8 calories per minute.

To calculate the total calories burned, simply multiply the activity's duration by its respective burn rate. For example, if the athlete runs for \(x\) minutes, they burn \(10x\) calories. Similarly, lifting weights for \(y\) minutes burns \(8y\) calories.
Summing these values provides the total caloric output:\[ z = 10x + 8y \] This makes it easy to adjust the workout to meet specific calorie targets by manipulating the time spent on each activity.

Linear equations are equations of the first degree, meaning they graph as straight lines when plotted. Our objective function, \[ z = f(x, y) = 10x + 8y \], is a linear equation. It consists of two variables, each multiplied by a constant coefficient, and no variables are raised to a power higher than one.
Linear equations are critical in optimization problems because they simplify the computation of different scenarios. Here, by adjusting \(x\) and \(y\), we can directly see how the total calories (\(z\)) change.
This straightforward relationship shows why linear equations are often used when modeling real-world scenarios, such as in nutrition and exercise planning, ensuring clarity and predictability in outcomes.