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Applications of Linear Functions and Slope
Linear functions apply to real world problems that involve a constant rate.
Learning Objective

Interpret constant rate of changes problems as applications of the slope of a linear function
Key Points
 If you know a realworld problem is roughly linear, such as the distance you travel when you go for a jog, all you need is two points and you can graph the function and make some assumptions as to what happens beyond the two points, so long as you maintain the same rate.
 The slope of a function is the same as the rate of change for the dependent variable. For instance, if you're graphing distance vs. time, then the slope is how fast your distance is changing with time, or in other words, your speed.
 When checking where two linear functions intersect, set them equal to each other and solve for the dependent variable, x.
Terms

linear equation
A polynomial equation of the first degree (such as x = 2y  7).

slope
The ratio of the vertical and horizontal distances between two points on a line; zero if the line is horizontal, undefined if it is vertical.
Full Text
Let's say that one day you decide to start training for a marathon. You start at your house, stretch, and look at your watch. It reads 6:00 pm. You plug in your headphones and begin to run around town. After a while, you realize you can't run anymore, and look at your watch. It reads 7:30 pm, and you're 7.5 miles from home. How fast was your average speed over the course of the run?
Our two variables are time and distance, and you have the data for two separate points. The first point is at your house, where your watch read 6:00, let's call this the beginning time and set it to 0. So our first point is (0, 0) as we hadn't run any distance yet. Let's think about our time in hours. So our second point is 1.5 hours later, and we ran 7.5 miles. So our second point is (1.5, 7.5). Our speed is simply the slope of the line connecting the two points. The slope, given by
becomes
To graph this line, we need the yintercept and the slope. We have just calculated the slope, but what's the yintercept? Since we started at (0, 0), we can see that the yintercept is 0. So our final function is
With this new function, we can now answer some more questions.
 How many miles had we run after the first half hour?
 If we kept running at the same pace for a total of 3 hours, how many miles will we have run?
$26=5x$ , so$x=\frac{26}{5}=5.2$ hours.
There are many such applications for linear equations. Anything that involves a constant rate of change can be nicely represented with a line with the slope. Indeed, so long as you have just two points, if you know the function is linear, you can graph it and begin asking questions! Just make sure what you're asking and graphing makes sense. For instance, in the marathon example, the domain is really only
Let's look at another application. Two trains start 200 miles apart. They are travelling towards each other. The first train is travelling 40 miles per hour, while the second train is travelling 60 miles per hour. When and where do they meet? For simplicity we should put one of the trains starting at (0, 0), and so the other train must start at either (0, 200) or (0, 200) as it is 200 miles away. We'll choose (0, 200). The first train is travelling 40 miles per hour towards the other train, which is in the positive y direction, so the slope is positive 40. The second train is travelling towards the first at 60 miles per hour, which is in the negative y direction, so the slope is negative 60. Now we have the equations for both of the functions. Let

$40x = 60x + 200$ 
$100x = 200$ 
$x = 2$
They meet after 2 hours of travel. To find out where they meet, plug x=2 into one of the equations, say f(x). Thus
Trains Used in Applications
Trains are just one example of things that can be used with linear functions. For instance, to see when two trains travelling at constant rates towards each other meet is a simple linear function.
Assign just this concept or entire chapters to your class for free.
Key Term Reference
 constant
 Appears in these related concepts: Inverse Variation, Quadratic Functions of the Form f(x) = a(xh)^2 + k, and Direct Variation
 dependent variable
 Appears in these related concepts: Formulating the Hypothesis, Converting between Exponential and Logarithmic Equations, and Experimental Research
 distance
 Appears in these related concepts: Inequalities with Absolute Value, Symmetry, and The Distance Formula and Midpoints of Segments
 domain
 Appears in these related concepts: Restricting Domains, Phenotypic Analysis, and Classification of Prokaryotes
 equation
 Appears in these related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 function
 Appears in these related concepts: Inverse Functions, Average Value of a Function, and Functions and Their Notation
 graph
 Appears in these related concepts: Reading Points on a Graph, Graphing Equations, and Graphing Functions
 linear
 Appears in these related concepts: Exponential Growth and Decay, Graphs of Linear Inequalities, and Delivery Tips
 linear function
 Appears in these related concepts: Zeroes of Linear Functions, Linear Inequalities, and The Linear Function f(x) = mx + b and Slope
 point
 Appears in these related concepts: The Intermediate Value Theorem, Quadratic Functions of the Form f(x) = ax^2 + bx + c, Where a is not Equal to 0, and Relative Minima and Maxima
 set
 Appears in these related concepts: Introduction to Sequences, Expressions and Sets of Numbers, and Executive Function and Control
 variable
 Appears in these related concepts: Calculating the NPV, Fundamentals of Statistics, and Math Review
 yintercept
 Appears in these related concepts: Circles, Basics of Graphing Polynomial Functions, and SlopeIntercept Equations
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Source: Boundless. “Applications of Linear Functions and Slope.” Boundless Algebra. Boundless, 21 Jul. 2015. Retrieved 10 Feb. 2016 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/graphsfunctionsandmodels2/functionsanintroduction17/applicationsoflinearfunctionsandslope1035009/