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5.1: Corequisite- Graphing Points, Linear Equations and Models

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    148614
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    SPECIFIC OBJECTIVES

    By the end of this lesson, you should understand that

    • addition and subtraction are inverse operations.
    • multiplication and division are inverse operations.
    • solving for a variable includes isolating it by “undoing” the actions to it.
    • multivariable models have one output variable and more than one input variable.

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

    • solve for a variable in a linear equation.
    • explicitly write out the order of operations to evaluate a given equation.

    PROBLEM SITUATION: CALCULATING BLOOD ALCOHOL CONTENT

    Blood alcohol content (BAC) is a measurement of how much alcohol is in someone’s blood. It is usually measured as a percentage. So, a BAC of 0.3% is three-tenths of 1%. That is, there are 3 grams of alcohol for every 1,000 grams of blood. A BAC of 0.05% impairs reasoning and the ability to concentrate. A BAC of 0.30% can lead to shortness of breath, a blackout, and loss of bladder control. In most states, the legal limit for driving is a BAC of 0.08%.1

    BAC is usually determined by a Breathalyzer, urinalysis, or blood test. However, the Swedish physician E.M.P. Widmark developed the following equation for estimating an individual’s BAC.2 This formula is widely used by forensic scientists:

    \[B = -0.015\cdot t + \left( \dfrac{2.84\cdot N}{W\cdot r}\right) \nonumber\]

    B = percentage of BAC

    N = number of “standard drinks” (A standard drink is one 12-ounce beer, one 5-ounce glass of wine, or one 1.5-ounce shot of liquor.) N should be at least one.

    W = weight in pounds

    r = the distribution rate for alcohol through the body (this value is a constant), 0.68 for males

    and 0.55 for females

    t = number of hours since the first drink

    (1) Looking at the equation, discuss why the items (t, N, W, and r) on the right of the equation make sense in calculating BAC.

    (2) Consider the case of a male student who has three beers and weighs 120 pounds. Simplify the equation as much as possible for this case. What variables are still unknown in the equation? Round values (when necessary) to the nearest thousandth.

    (3) (a) Using your simplified equation from Question 2, find the estimated BAC for this student one hour, three hours, and five hours after his first drink.

    (b) What patterns do you notice in the data from Question 3 (a)?

    (4) How did you arrive at the BAC values mathematically? For example, did you multiply, add, subtract, etc., and what did you do first? Outline the steps that you took to get the answers for Question 3(a).

    (5) How long will it take for this student’s BAC to be 0.08, the legal limit? How long will it take for the alcohol to be completely metabolized resulting in a BAC of 0.0? Round to one decimal place.

    ALGEBRAIC EXPRESSIONS

    In algebra, a term is an individual part of an algebraic expression. Terms are separated by addition (+) or subtraction (–) signs, and can consist of numbers, variables (letters), or the product of numbers and one or more variables. In instances where a number and variables are being multiplied, the number is called a coefficient. For example, the expression below has three terms as shown in boxes. Notice that the last term is (–5). In the original expression, this was written as minus 5, but this can be rewritten as adding negative 5 as shown below. When breaking an expression into terms, you ask, what is being added?

    Equation minus a cubed plus two b minus 5.

    Terms Coefficients
    –a3 –1
    2b 2
    –5 Does not have a coefficient because there is no variable. This is called a constant term because it never changes.

    (6) State the number of terms in each expression:

    (a) 3x + 4

    (b) 5x – 4x2 + 2

    (c) 5

    (7) What is the coefficient of the x2 term in the expression 5x – 4x2 + 2 ?

    GRAPHING REVIEW

    This section will help you review some graphing skills that will be needed in many future lessons. In upcoming lessons, you will make graphs on a coordinate plane like the one shown below. You will learn about using a coordinate plane and practice how to graph points.

    Coordinate plane graph with axes ranged from -4 to 4. The origin (0,0) is labeled, and point A-J are marked.

    We will begin with some vocabulary. A coordinate plane has two axes that measure distance in two dimensions. The horizontal axis goes from left to right. In previous classes, you may have called this the x-axis. The vertical axis goes up and down. This is sometimes called the y-axis. The axes are two number lines that create a grid on the coordinate plane. (Note: Axis is singular and axes is plural.)

    The point at which the two axes intersect or cross is called the origin. This point represents 0 for both axes. To the left of this point, the horizontal axis is negative; to the right it is positive. Below the origin, the vertical axis is negative; above the origin it is positive. You can see this in the numbers along each axis above. These numbers are called the scale.

    Each location or point on the coordinate plane is defined by an ordered pair. You can think of this as the address of a point. An ordered pair must contain two numbers. Ordered pairs are written in a set of parentheses ( ) with a comma separating the numbers, such as (2, 4). The first number is the distance and direction going left or right from the origin and the second number is the distance and direction going up or down. The ordered pair for the origin is (0, 0).

    Follow these steps to find the point represented by the ordered pair (2, 3):

    1. First, think about the “address” of the point. If this were a street address, the ordered pair tells you to walk 2 blocks horizontally in the positive direction (right) and then walk 3 units vertically in the positive direction (up).
    2. Start at the origin. Go 2 units to the right because this is the positive side of the horizontal axis.
    3. Go 3 units up.

    Point A on the graph above is the point (2, 3). A few other examples from the graph are given below:

    Point B: (–3, 1)

    Point E: (0, 1)

    Point F: (4, 0)

    (8) Write the ordered pairs for the following points on the graph.

    Point C:

    Point D:

    Point G:

    Point H:

    Point I:

    Point J:

    Coordinate plane graph with axes ranged from -4 to 4. The origin (0,0) is labeled, and point A-J are marked.

    _______________________________

    1 http://en.Wikipedia.org/wiki/Blood_alcohol_content

    2 http://www.avvo.com/legal-guides/ugc/explaining-blood-alcohol-levels-bac-what-does-the-08-bac-mean-how-many-drinks-is-that-1


    This page titled 5.1: Corequisite- Graphing Points, Linear Equations and Models 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.