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Mathematics LibreTexts

2.1: Lines

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    1. Lines (definitions)

    Everyone knows what a line is, but providing a rigorous definition proves to be a challenge.

    Definition: Line

    A line with slope \(m\) through a point \(P = (a,b)\) is the set of all points \((x,y)\) such that

    \[\dfrac{y-b}{x-a}= m.\]

    2. The Slope Intercept Form of the equation of a Line

    Given a point \((x_1,y_1)\) and a slope \(m\), the equation of the line is

    Definition: Slope Intercept Equation of a Line

    \[y-y_1=m(x-x_1)\]

    3. Piecewise Linear Functions

    A function is piecewise linear if it is made up of parts of lines

    Example 1

    \[f(x)=\begin{cases} x+4 & \text{if }x\leq-2 \\ 2x-1 & \text{if } -2<x<1 \\ -2x & \text{if } x\geq1\end{cases}\]

    We graph this line by sketching the appropriate parts of each line on the same graph.

    4. Applications

    Example 2

    Suppose you own a hotel that has 150 rooms.  At $80 per room, you have 140 rooms occupied and for every $5 increase in price you expect to have two additional vacancies.  Come up with an equation that gives rooms occupied as a function of price.

    Solution

    Let \(x\) be the price of a room and \(y\) be the number of rooms occupied.  Then we have an equation of a line that passes through the point \((80,140)\) and has slope \(-\frac{1}{5}\).  Hence the equation is:

    \[y - 140 = -\dfrac{1}{5}(x - 80)\]

    or 

    \[y = -\dfrac{1}{5} x + 16 + 140\]

    or

    \[y = -\dfrac{1}{5} x + 156.\]

    Exercise 1

    What should you do if your two year old daughter has a 40 degree C temperature?  

    Hint:  We have the two points:  \((0,32)\) and \((100,212)\).

    Exercise 2

    Suppose that your company earned $30,000 five years ago and $35,000 three years ago.  Assuming a linear growth model, how much will it earn this year?

    Exercise 3

    My rental was bought for $204,000 three years ago.  Depreciation is set so that the house depreciates linearly to zero in twenty years from the purchase of the house.  If I plan to sell the house in twelve years for $250,000 and capital gains taxes are 28% of the difference between the purchase price and the depreciated value, what will my taxes be?

    Exercise 4

    Wasabi restaurant must pay either a flat rate of $400 for rent or 5% of the revenue, whichever is larger.  Come up with the equation of the function that relates rent as a function of revenue

    Larry Green (Lake Tahoe Community College)