In Exercises 32 and 33, use the following information. The...

In Exercises 32 and 33, use the following information. The graph below represents the need for low-income rental housing in the United States and the number of affordable rental units available. Use the points (0, 6200) and (25, 10,500) to write an equation for the number of housing units needed. Then use the points (0, 6500) and (25, 6100) to write an equation for the number of affordable units available.

Key Concepts

Two-Point Formula

The two-point formula is used to determine the slope of a line when two points on the line are known. By calculating the differences in the y-values and the x-values between the two points, the formula m = (y2 - y1) / (x2 - x1) provides the slope, which can then be used to establish the entire linear equation.

Slope-Intercept Form

The slope-intercept form, y = mx + b, is a common way of writing linear equations. This form highlights the slope (m) and the y-intercept (b), where the y-intercept represents the value of y when the independent variable x is zero. It is especially useful for quickly graphing linear relationships and interpreting their characteristics.

Modeling with Linear Equations

Modeling with linear equations involves using mathematical expressions to represent real-world situations, where changes between variables are constant. This technique is valuable for making predictions, understanding trends, and analyzing relationships in contexts like economics, science, and social studies.

Linear Equations

Linear equations are mathematical expressions that show a straight-line relationship between two variables. They are often written in the form y = mx + b, where m is the slope and b is the y-intercept, making it straightforward to calculate and interpret changes in the variables.

Slope

The slope of a line represents the rate at which the dependent variable changes relative to the independent variable. It is calculated as the difference in the y-values divided by the difference in the x-values between any two points on the line, and it indicates both the direction and steepness of the line.

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