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5.2: Corequisite- Solving Linear Equations, Slope

  • Page ID
    148616
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    SPECIFIC OBJECTIVES

    By the end of this lesson, you should understand that

    • linear models are appropriate when the situation has a constant increase/decrease.
    • slope is the rate of change.
    • the rate of change (slope) has units in context.
    • different representations of a linear model can be used interchangeably.

    By the end of this lesson, you should be able to

    • label units on variables used in a linear model.
    • make a linear model when given data or information in context.
    • make a graphical representation of a linear model.
    • make a table of values based on a linear relationship.
    • identify and interpret the vertical intercept in context.

    REPRESENTATIONS OF MODELS

    In Module M, you will learn about how different models (such as equations in context) can be useful in examining some situations encountered in real life. A model is a mathematical description of an authentic situation. You can also say that the mathematical description “models” the situation. You will practice using four representations to express the models for situations. We will start the module with the more familiar linear models and then expand to other models.

    Verbal Description (Words)

    A verbal description explains the relationship in words. Some relationships are difficult to put into words, while in other cases a verbal description can help you make sense of what the relationship means in the context.

    Numerical Representation (Table)

    Another way that you could represent a relationship is in a table. Tables are helpful for recognizing patterns and general relationships or for giving information about specific values. A table should always have labels for each column. The labels should also include units when appropriate.

    Graphical Representation (Graph)

    A graph provides a visual representation of the situation. It helps you see how the variables are related to each other. It also helps you to make predictions about future values as well as values in between those in your table. The horizontal and vertical axis of the graph should be labeled, including units.

    Algebraic Model (Equation)

    You will consider how to write a mathematical equation for a relationship. In writing equations, it is always important to define what the variables represent, including units.

    Summary

    Throughout this course, you have learned that having the skill to move between different forms and tools is important in problem solving. Alternating among the four representations of mathematical relationships is another example of this. In some cases, you may struggle with writing an equation, but find it helpful to start with a table. You might want a graph for a visual representation, but also need to express a relationship in words. It is important that you can translate one form into another and that you can choose which form is most useful in a specific situation.

    PROBLEM SITUATION: DAILY LATTE

    A local coffee shop offers a coffee card that you can preload with any amount of money and use like a debit card each day to purchase coffee. At the beginning of the month (when you get your paycheck), you load it with $50. Your favorite small soy latte costs $2.63.

    (1) Pretend you purchase a latte every weekday. Estimate whether the amount of money you loaded onto your coffee card will last until the end of the month.

    (2) Fill in the table and graph below to help you answer the question: Will your $50 coffee card last until the end of the month? (Note: Any given month has about 22 weekdays; there could be as few as 20 weekdays in February, or as many as 23 in some other months.)

    (a) Fill in the table.

    Lattes (n) Amount Remaining on Card (A)
    0 50
    1 50 – 2.63
    2  
    3  
    4  
    10  
    15  
    20  
    23  

    (b) Graph the data from the table in (a). Be sure to include scales and labels on the axes. Let the horizontal axis be the number of lattes purchased and the vertical axis be the amount of money remaining on the card. Note: If completing this problem online, follow the instructions given online to create your graph.

    Blank coordinate plane quadrant I graph.

    (c) Build a linear model. Use A to represent the remaining money on the coffee card and n to represent the number of lattes purchased.

    (d) Answer the following questions:

    (i) Will the card last the entire month?

    (ii) If so, how much money is left at the end of the month?

    (iii) If not, how many days will it last?

    The two mathematical relationships in the plans are linear. They have certain important characteristics. The following terms are vocabulary words you need to know to talk about linear models and other types of mathematical relationships. You will discuss these in your group. Make sure you take good notes about what each means.

    Constant rate of change: This is the characteristic that defines a linear relationship. A constant change can be seen both with the table and the graph.

    Slope: This is a ratio that describes the rate of change. Be sure you understand the slope and its relation to the graph as the rise (vertical movement) of the line over the run (horizontal movement). The concept of slope is the rate of change of the quantity represented. The slope is multiplied by the (input) variable because you use it as a rate to find a new value.

    Intercepts: The point at which a graph touches one of the axes. There is a vertical intercept and a horizontal intercept.

    Vertical intercept: The vertical intercept is sometimes called the starting value. It is the point at which the graph touches the vertical axis. Note that this often has important meanings in context.

    Horizontal intercept: The point at which the graph intersects the horizontal axis, where the value of the output is zero.

    Linear Model: A = b + mt. This can also be written as A = mt + b. In either case, A is the amount of the quantity at any time t; m is the constant rate of change (the slope); and b is the initial amount (when t = 0).

    (3) (a) What is the rate of change for the model above in Problem Situation: Daily Latte? What does it mean in the context of the problem situation? Remember to include units.

    (b) What is the vertical intercept for the model above in Problem Situation: Daily Latte? What does it mean in the context of the problem situation?

    (c) Does the model above in Problem Situation: Daily Latte have a horizontal intercept? If so, what is it and what does it mean in the context of the problem situation?


    This page titled 5.2: Corequisite- Solving Linear Equations, Slope is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Carnegie Math Pathways (WestEd) via source content that was edited to the style and standards of the LibreTexts platform.