Question

How can linear functions be used to model situations and solve problems?

Answer 1

Linear functions, represented by the equation y = mx + b, model the relationship between two variables and can be used to make predictions and solve problems by determining how one variable affects another.

Step-by-step explanation:

Linear functions can be powerful tools in modeling situations and solving a wide range of problems. These functions are typically represented by the algebraic equation y = mx + b, where 'm' represents the slope and 'b' indicates the y-intercept. The slope describes the rate of change between the variables, indicating how much y will change as x increases by one unit. The y-intercept is the value of y when x is zero, essentially where the line crosses the y-axis on a graph.

For instance, in modeling the relationship between two variables, such as the number of flu cases over a period of years, we can set the year as the independent variable x and the number of flu cases as the dependent variable y. By finding a linear pattern in historical data, we can establish a linear equation that describes this relationship, enabling predictions for future years.

As another example, consider the equation y = 3,000x + 500 which models the total cost of attending a college (y) based on the number of years enrolled (x). Here, the slope 3,000 indicates the additional cost per year, and 500 could be a fixed cost that occurs regardless of the number of years enrolled.

Answer 2

One day, you go to a store and buy a pencil. It costs $0.25.
You notice the unit price $0.25/pencil.

The next day you buy 3 pencils, they cost $0.75.
You notice the unit price is $0.75/(3 pencils) = $0.25/pencil.

For each extra pencil you buy, the increase in price is always $0.25.
This suggests a linear relation between the number of pencils, p, and the cost of the pencils, c.

You use a linear equation to give you the cost of pencils based on the number of pencils.

c = 0.25p

One day, you happen to have $4.50 in your pocket.
You'd like to buy 15 pencils, but you are not sure you have enough money.
You use your handy linear equation, and you replace p, the number of pencils, with 15 to find their cost, c.

c = 0.25p

c = 0.25 * 15

c = 3.75

After using your equation, you see that 15 pencils cost $3.75, and since you have $4.50 you have enough money to buy them. You're very happy about this, and, as soon as you get home, you write this experience in your journal, using one of the new pencils you just bought, and you make it a point to include the linear equation that helped you.