In this explainer, we will learn how to model real-world examples on linear equations.
In many real-life situations, we are interested in comparing quantities to each other and the underlying relationships they might have. Ideally, we want to be able to create mathematical models that accurately describe these relationships, since this allows us to understand their behavior better and gives us a way to predict future behaviors. We cannot always expect real-life data to be modeled easily by simple mathematical equations, but it is useful to be aware of basic patterns that can arise so that we can spot them when they occur.
One of the most common kinds of relations is that of proportionality. We can define this as follows.
Two quantities and are proportional to each other if they vary at the same rate, that is, if for a constant . It is also common to use the phrase “directly proportional” and to denote this relation by .
Graphically speaking, we recall that equations of the form are straight lines, where the value of dictates the slope of the line.
We can see that for the line , if we start at the point , we can multiply both variables by 3 to get us another point on the line, . Alternatively, multiplying them both by gets us a third point on the line, . In general, this is a property of any line of the form . That is, if we multiply both variables by the same value, then we obtain another point on the line.
We also observe that all of the lines above pass through the origin . This is to be expected, since for a line of the form , the -intercept is at . Thus, since in this case, the -intercept is at .
Proportional quantities appear in many real-life situations. For instance, the force used on a spring is proportional to the distance , it is extended:
We can see that the two quantities vary at the same rate in the sense that the stronger the force used the further the spring is stretched.
A more everyday example is that of price. If something has a fixed price, then the more of it we buy, the more money we have to pay. In this way, we can say that quantity and total cost are proportional to each other.
Let us go through an example where we can test if two quantities are proportional to each other by considering their graph.
Example 1: Using a Graph to Determine Whether a Relationship Is Proportional
The graph shows the relationship between the radius and the circumference of a circle in centimetres. Are the variables proportional?
In this example, we have been given a graph comparing the radius of a circle to its circumference and we need to determine whether the two measurements are proportional.
We may already be aware of the formula that calculates the circumference using the radius, which is
Recalling that two variables are proportional if they are related by an equation of the form , where is a constant, we can see that if is the circumference, is equal to , and is the radius, then the equation is of the correct form, so the variables must be proportional.
Nevertheless, this example can be answered even if we have no prior information about the variables. We can say that a line must be of the form if the following qualities hold true:
- The graph passes through the point .
- The graph is a straight line.
Since these are both true, we can say that the relationship is proportional.
So far, we have only covered proportionality as a form of relationship between quantities, but this is, in fact, a specific example of a linear relationship, meaning a relationship that can be described using any straight line. We can more precisely define a linear model as follows.
Definition: Linear Model
A model of the relationship between two variables and is linear if it is of the form where and are constants.
We note that the key difference between a proportional relationship and a linear one is that a linear relationship has the additional term, meaning the -intercept does not necessarily have to be at 0. As a result, two variables in a linear relationship do not necessarily have the property that scaling one by a constant implies the same scaling of the other. Indeed, this is only the case when (i.e., when they are directly proportional).
In this explainer, we want to be able to approximate real-life relationships using linear models. When we approximate things, we cannot always expect the real-life quantities to correspond exactly to our mathematical model of them. For this reason, even if our data does not exactly fit a linear model, we can still conclude that a linear model is appropriate if it is “close” to the model. This concept of being close is not one that can be easily quantified; instead, we have to draw our conclusions by looking at the data and making reasonable assumptions. The most important thing to keep in mind is that the further the data lies from the straight line that models that data, the less appropriate that model is.
Let us consider an example where we must decide whether a linear model is appropriate for a given data set.
Example 2: Using a Table to Determine Whether a Relationship Is Approximately Linear
A hiker is walking along a mountain trail and records the distance they have walked at various time intervals, as shown below.
By drawing a graph, determine whether the data can be approximated by a linear model.
We recall that a linear model is one that forms a straight line. As the question suggests, we can approach this example by drawing a graph and confirming visually whether a straight line is an accurate approximation. Doing this, we get the following graph.
At a glance, the points do appear to mostly be in a line. To confirm this, we can draw a line through the first two points and extend it on either side. If we do this, we get the following.
We can see that almost all of the points lie on the line and one point is very close to the line. Even though the data is not exactly linear, as an approximation, we can say that a linear model is indeed appropriate. So, the answer is yes.
As of yet, we have not had to explicitly find the equations describing any linear models. This is a useful skill for us to have, since the constants in the equation of a line have their own meanings that can be interpreted. For instance, let us recall the slope–intercept form of a line:
Here, is the slope, which tells us that each increase of the quantity by one unit corresponds to an increase in by . In other words, it gives us the rate of change of with respect to . On the other hand, gives us the -intercept, which tells us the value of when is equal to 0. This can be significant in terms of the initial values of the problem.
Sometimes, we will be asked questions where we need to interpret the meanings of these constants in a real-world context, which will require us to analyze the information given to us and relate it to the above qualities of and . Let us see an example of this.
Example 3: Finding the Equation Linking Two Variables given a Graph and Analyzing the Equation
Farida collects identical samples, places them in a dish, and weighs the dish on a scale. She plots the weight for various numbers of samples.
- Write an equation linking the weight to the number of samples in the form .
- What is the physical significance of the value of ?
To find the equation of a line from its graph, recall that we can find its slope and its -intercept and substitute these into the slope–intercept form of the line (i.e., ).
The slope can be calculated by picking any two points on the line and and using the equation for the slope of a line:
Let us choose and . Substituting these coordinates and letting our slope be , we get
Next, we can find the -intercept by identifying where the line intersects the . Since this is at the point , we can say that the -intercept is .
Putting this together, we have
To understand the physical significance of (which is 4 in the above equation), we should interpret the information given to us in the question. The samples are placed in a dish and then weighed, which implies that the dish itself has weight. Considering the graph, we can see that when there are 0 samples on the tray, the weight is 4 g.
Thus, is the weight of the dish in grams.
As shown in the previous example, finding the equation of the line that models a relationship can be useful, but we do not necessarily need to go that far to interpret the real-life meaning of data. In the next example, we will demonstrate how we can find information directly from the graph.
Example 4: Finding How Much Fuel Is Used by a Car Based on a Linear Model
The graph below represents a relation between the time in hours and the amount of fuel in a car tank in litres. Determine the amount of fuel remaining after 15 hours of driving and the amount of fuel consumed per hour.
For the first part of this problem, we need to determine the amount of fuel remaining after 15 hours of driving. To do this, we want to find the value of (the fuel) at the point when (the time in hours) is equal to 15. Graphically, we can do this by drawing a line from the to the graph and from the graph to the .
Thus, we find that at this point, meaning 24 litres of fuel remain in the tank at this point.
Next, we need to find the amount of fuel consumed per hour. This can be found by considering the slope of the line, since this is the rate that fuel changes with respect to time. To do this, we can recall the formula for the slope of a line given two points and on it:
For simplicity, we will take the points at the ends of the line, and , giving us the following slope:
This number signifies that for every increase in by 1, will decrease by . Translating this to real-world terms, we can say that litres of fuel are consumed per hour.
For our final example, we will consider how it is possible to predict future behavior by considering a linear model that approximates it.
Example 5: Estimating the Depth of a Diver Falling at a Linear Rate
A scuba diver is ascending from their dive at the rate shown in the diagram.
- Assuming they continue at the same rate, how much time will have passed when they reach the surface?
- In the last part of the ascent, they decide to slow their rate of ascent to 5 metres per minute. Given that this is the case, how much time will have passed when they reach the surface?
To understand this question, we have to understand what the rate refers to in the context of the diagram. Since the two variables involved are the depth in metres and the time in minutes, the rate refers to how many metres are ascended per minute.
Recall that we can calculate the rate of change of two quantities with respect to each other by finding the slope of the line that models them. Given two points and on the line, the formula for the slope is
If we take the points and on the line and substitute them into the formula, we get
Thus, for every increase in time by 1 minute, the depth decreases by 9 metres, or, to put it another way, the diver ascends at a rate of 9 metres per minute. Since the diver is at a depth of 9 metres at the 3-minute mark, this means that at the current rate, an extra minute will be required, giving us a total of 4 minutes to reach the surface. We highlight this approximation below using a dotted line.
So, as we can see, we can also obtain this answer by continuing the line until it reaches the .
Now, in the second part, we find that the diver decides to slow their rate of ascent down to 5 metres per minute. Since there are 9 metres remaining, we can find the time this will take by dividing 9 by 5. That is, the remaining time taken will be
Adding this time to the initial 3 minutes, we see that the total time will be 4 minutes and 48 seconds. We can plot this on the graph at the point .
We note that this would result in the model not being linear, since the slope would not be the same throughout the line.
In conclusion, the initial predicted time is 4 minutes and the updated time is 4 minutes and 48 seconds.
Let us finish by considering the key points of this explainer.
- If two quantities and are proportional, then they vary at the same rate and they can be related by where is a constant.
- The graph of a proportional relationship is a straight line passing through the origin.
- If two quantities and obey a linear model, they are related by where and are constants.
- The graph of a linear model is any straight line.
- A linear model can approximate a real-life problem even if the data does not exactly fit it, as long as the data is close to the line.
- We can answer real-world problems approximated by linear models by considering the graph; in particular, we can examine the points that lie on the line, its slope, and its intercepts with the axes.