A real estate office handles an apartment complex with 50...

A real estate office handles an apartment complex with 50 units. When the rent per unit is $\$ 580$ per month, all 50 units are occupied. However, when the rent is $\$ 625$ per month, the average number of occupied units drops to $47 .$ Assume that the relationship between the monthly rent $p$ and the demand $x$ is linear. (a) Write the equation of the line giving the demand $x$ in terms of the rent $p$ . (b) Use a graphing utility to graph the demand equation and use the trace feature to estimate the number of units occupied when the rent is $\$ 655 .$ Verify your answer algebraically. (c) Use the demand equation to predict the number of units occupied when the rent is lowered to $\$ 595 .$ Verify your answer graphically.

A real estate office handles an apartment complex with 50 units. When the rent per unit is $\$ 580$ per month, all 50 units are occupied. However, when the rent is $\$ 625$ per month, the average number of occupied units drops to $47 .$ Assume that the relationship between the monthly rent $p$ and the demand $x$ is linear.
(a) Write the equation of the line giving the demand $x$ in terms of the rent $p$ .
(b) Use a graphing utility to graph the demand equation and use the trace feature to estimate the number of units occupied when the rent is $\$ 655 .$ Verify your answer algebraically.
(c) Use the demand equation to predict the number of units occupied when the rent is lowered to $\$ 595 .$ Verify your answer graphically.

Key Concepts

Linear Equations

Linear equations describe relationships where the change between two variables is constant. They are expressed in the form y = mx + b, where m is the slope (indicating the rate of change) and b is the y-intercept. This foundational concept is critical for modeling and predicting outcomes in various scenarios by establishing a straight-line relationship between quantities.

Linear Models in Economics

Linear models are used in economics to represent relationships such as between price and demand. By assuming a constant rate of change, these models simplify complex market behaviors, allowing for the prediction of outcomes (like demand for an apartment) based on changes in price. This approach provides an accessible method for making forecasts and analyzing economic trends.

Slope and Rate of Change

The slope in a linear equation represents the rate at which one variable changes with respect to another. In the context of modeling, it quantifies how much the dependent variable (like demand) is expected to change when the independent variable (such as rent) increases by one unit. This concept is essential for understanding and predicting the behavior of the linear relationship.

Graphical Verification

Graphical verification involves plotting the linear equation on a coordinate system to visually confirm the relationship between variables. Graphing utilities and features like trace allow users to identify specific values and trends, providing a practical method to verify algebraic solutions. This reinforces understanding by linking numerical results with their visual representation.

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