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Writing & Evaluating Real-Life Linear Models: Process & Examples

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  • 0:01 A Real Life Dilemma
  • 0:37 Linear Model Defined
  • 1:53 Independent and…
  • 2:58 Writing Linear Equations
  • 4:54 Evaluating Linear…
  • 9:50 Lesson Summary
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Lesson Transcript
Instructor: Cathryn Jackson

Cat has taught a variety of subjects, including communications, mathematics, and technology. Cat has a master's degree in education and is currently working on her Ph.D.

You make decisions about budgeting and other financial issues using linear models without even realizing it. Learn how to write and evaluate linear models.

A Real Life Dilemma

Jill is putting together her monthly budget. Right now, she is working on her electric bill. She has a statement from her electric company that shows the kilowatts per hour that she uses for each month. She knows the rate per kilowatt per hour is 8.2 cents, but she doesn't know how much it costs her per month. How is Jill going to estimate her monthly bills if she doesn't know how much her electric bills cost her? Jill can use a linear model to write equations that will help her predict the cost of her monthly bills.

Linear Model Defined

A linear model is a comparison of two values, usually x and y, and the consistent change between the values. In the opening story, Jill was analyzing two values: the amount of electricity used and the total cost of her bill. The change between these two values is the cost of each kilowatt hour. Still a little confused? We'll come back to this later.

For right now, let's talk about that consistent change, which is also known as the rate of change; in algebra, we call that the slope. You're probably familiar with the slope of a line: it describes both the length and the steepness of the line.

Okay, so back to our example. You've probably already figured out that the more electricity Jill uses, the higher her bill will be at the end of the month. Therefore, the rate of change shows us just how much that bill will increase as Jill uses more electricity.

As you can see, linear models are very applicable to real life and can be used to predict certain information that is useful for making decisions and problem solving.

Independent and Dependent Variables

Before we can figure out how much Jill's bill is going to be, we need to know the independent and dependent variables.

An independent variable is a variable that stands alone and is the input for the equation. For example, if you were to conduct an experiment and determine how much water a plant needs to survive, you might try using different amounts of water on plants in the same lighting and soil conditions. You are using water as the independent variable, or the input in this situation. In a linear equation, x represents the independent variable.

The dependent variable is the result of the independent variable, or the output. After watering your plants, you may want to measure how much each plant grew. The amount the plant grows is dependent upon how much water you gave it. In a linear equation, y represents the dependent variable.

Writing Linear Equations

In our example, Jill has two pieces of information that are very important to this problem: the amount of electricity and the cost of the bill. Which piece of information is the dependent variable and which piece of information is the independent variable? Well, the cost of the bill is dependent upon how much electricity Jill uses. Therefore, the cost of the bill is the dependent variable and the amount of electricity is the independent variable. We can write our linear model like this: y = .082x, where y is the cost of the bill, and x is the amount of electricity used.

You can use slope-intercept form, which is y = mx + b, to write equations for linear models. m is the slope or rate-of-change, and b is the y-intercept. Often, the y-intercept represents the starting point of the equation. For example, if Jill had a $20 per month fee for her electricity bill, then her bill would always be $20 or more. Therefore, our equation would be y = .082x + 20. The graph of our equation would look like this.

linear equation graph

Now Jill can use a graph like this to figure out how much each month's electricity bill will be. If she uses 1,200 Kilowatts per hour in September, then her bill would be about $118.40.

Evaluating Linear Models and Equations

Jill's husband, Danny, has a car that he wants to sell. The car is pretty average with little damage and average mileage. They have owned the car for five years. Danny wants to know how much his car will be worth now and in two years. He has developed the following formula to help in his decision making: y = -500x + 16,000. We can use what we know of linear models to understand and evaluate this equation.

Remember, the slope-intercept form of an equation is y = mx + b. We can see a similar pattern with Danny's equation: y = -500x + 16,000. First, let's examine the numbers in Danny's equation. The first number that appears in the equation is -500. This number is sitting in the m, or slope, spot in the slope-intercept equation. That tells me that -500 represents the rate of change in the scenario. That makes sense, because we know that Danny's car will likely depreciate in value as time passes. Therefore, the slope will be negative and go down, rather than up.

The next number that stands out is 16,000. This number is sitting in the b or y-intercept spot in the slope-intercept equation. That tells me that 16,000 represents the starting point for Danny's car. What kind of starting point? Well, since we are talking about cost and depreciation, I can guess that the car was originally worth $16,000.

Now, let's identify the independent and dependent variables. We already know that y represents the dependent variable, and x represents the independent variable, but what does that mean in regards to the scenario?

We already know that -500 represents rate of change in regards to time. Therefore, we can identify x as the year or time. This makes sense because x is the independent variable and time doesn't usually wait or depend on anything, much less the cost of a car.

That leaves us with y, or the dependent, variable. Since we already know that Danny wants to know the cost of the car after time, then we can deduce that y represents the depreciated cost of the car at a certain point in time.

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