Lesson Video: Modeling with Straight Lines Mathematics

Video Transcript

In this video, 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. We want to be able to create mathematical models that accurately describe these relationships, as this allows us to understand their behavior better as well as predict future behaviors. Of course, real-life data cannot always 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. Two quantities 𝑥 and 𝑦 are said to be proportional to each other if they vary at the same rate. We can express this as the equation 𝑦 equals 𝑘𝑥 for some constant value 𝑘. 𝑘 is known as the constant of proportionality. It is also common to describe such a relationship using the phrase “directly proportional,” which we can write using the notation shown.

Graphically speaking, we recall that equations of the form 𝑦 equals 𝑘𝑥 are straight lines passing through the origin, with the value of 𝑘 dictating the slope of the line. Taking any point on such a line and multiplying both its 𝑥- and 𝑦-coordinates by a constant will give the coordinates of another point on the same line.

Proportional quantities appear in many real-life situations. For instance, the force 𝐅 applied to a spring is proportional to the distance 𝑥 it is extended. The 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 an object has a fixed price, then the more of it we buy, the more money we have to pay. The cost and the quantity bought are proportional to one another. Let’s now consider our first example, in which we’ll test whether two quantities are in proportion by considering their graph.

The graph shows the relationship between the radius and the circumference of a circle in centimeters. Are the variables proportional?

We’ve been given the graph depicting the relationship between these two variables and asked to determine whether they are proportional. If two variables 𝑦 and 𝑥 are proportional, or are in direct proportion, then we recall that this means 𝑦 is equal to 𝑘𝑥 for some constant 𝑘. It follows that the graph of 𝑥 against 𝑦 will be a straight line passing through the origin, with a slope of 𝑘.

The graph we’ve been given is a straight line, and it does pass through the origin. So, we can deduce that the two variables are proportional. In fact, we know that the circumference of a circle is equal to 𝜋 times the diameter, or two 𝜋 times the radius. Hence, we have the equation 𝐶 equals two 𝜋𝑟, and the value of 𝑘, the constant of proportionality, is two 𝜋. Note that two 𝜋 is approximately equal to 6.3. And from the graph, we can confirm that the slope of the line is indeed equal to this value. So our answer is yes, the radius and circumference of a circle are proportional to one another.

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. More generally, a model of the relationship between two variables 𝑥 and 𝑦 is linear if it is of the form 𝑦 equals 𝑘𝑥 plus 𝑐, where 𝑘 and 𝑐 are both constants.

The key difference between a proportional relationship and a linear one is the addition of the plus 𝑐 term in the linear relationship, meaning the 𝑦-intercept does not necessarily have to be at zero. As a result, two variables in a linear relationship do not necessarily have the property that scaling one by a constant results in the same scaling of the other. This is only the case when 𝑐 equals zero and the variables are directly proportional.

In this video, we want 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. For this reason, even if our data does not exactly fit a linear model, we may conclude that a linear model is still appropriate if the fit is close enough, as we will see in our next example.

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’re told that we should approach this question by drawing a graph. So we’ll begin by setting up some axes that cover the ranges of both variables. We can then plot the five coordinate points on the graph. We’re asked whether the data can be approximated by a linear model, which means that the data forms a straight line.

To test this, we can draw a line through the first two points and extend it on either side. We can then see that four of the points lie exactly on the line. And the fifth point is very close to the line. This means that whilst the data isn’t exactly linear, it is very close. And therefore the data can be approximated by a linear model.

So far, we have not had to explicitly find the equations describing any linear models. This is a useful skill to have, since the constants in the equation of a straight line have particular meanings. Recall that we defined a linear model to be of the form 𝑦 equals 𝑘𝑥 plus 𝑐, where 𝑘 and 𝑐 are constants. This mirrors the slope–intercept form of the equation of a straight line, 𝑦 equals 𝑚𝑥 plus 𝑐, with 𝑘 and 𝑚 interchanged. Here, 𝑚 represents the slope of the line, which tells us that each increase of the quantity 𝑥 by one unit corresponds to an increase in 𝑦 by 𝑚 units. In other words, it gives us the rate of change of 𝑦 with respect to 𝑥. The value of 𝑐 gives the 𝑦-intercept of the line, which tells us the value of 𝑦 when 𝑥 is zero. This can be significant in terms of the initial values of a problem.

Sometimes, we will be asked questions where we need to interpret the meanings of these constants in a real-world context. Let’s consider an example of this.

Farida collects identical samples, places them in a dish, and weighs the dish on a scale. She plots the weights for various numbers of samples. Write an equation linking the weight 𝑊 to the number of samples 𝑁 in the form 𝑊 equals 𝑎𝑁 plus 𝑏. What is the physical significance of the value of 𝑏?

The first part of this question asks us to write an equation linking the weight 𝑊 and the number of samples 𝑁. We can see from the graph and from the given format that this is a linear equation. In this form, the value of 𝑎 represents the slope of the line and the value of 𝑏 represents the 𝑦-intercept. We can determine both of these values from the graph.

The graph intercepts the 𝑦-axis at four. So this is the value of 𝑏. Considering the endpoints of the graph, the line travels five units to the right and four units up. Hence, the slope of the line, which is its change in 𝑦 over its change in 𝑥, is four over five. And this is the value of 𝑎. Substituting these values into the equation, we have that the equation linking the weight and the number of samples is 𝑊 equals four-fifths 𝑁 plus four.

In the second part of the question, we are asked what the physical significance of the value of 𝑏 is. So we’re being asked to interpret this model. 𝑏 is the 𝑦-intercept of the straight line, which corresponds to an 𝑥-value of zero. This means that 𝑏 is the weight shown on the scale when there are no samples in the dish. Therefore, 𝑏 represents the weight of the dish itself, the units for which are grams.

In our final example, we will consider how it is possible to predict future behavior by considering a linear model that approximates a real-life situation.

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 five meters per minute. Given that this is the case, how much time will have passed when they reach the surface?

In this two-part question, we are asked to make two different assumptions. First, we assume that the diver continues to ascend at the same rate. Extending the line, we see that it will meet the 𝑥-axis at a value of four, indicating that when the depth is zero, or the diver is at the surface, the time that has passed will be four minutes. We can also see this from the slope of the graph. For every one minute that passes, the diver ascends nine meters. At the end of the graph, there are nine meters left to ascend. So, it will take a further one minute in addition to the three that have already passed.

In the second part of the question, we’re now asked to assume that the diver slows their rate of ascent to five meters per minute. As there are nine meters remaining, we can calculate the time taken by dividing this distance by the speed of five meters per minute. Nine over five minutes is equivalent to 1.8 minutes. We can calculate that this is equivalent to one minute and 48 seconds by multiplying 0.8 by 60. Adding the previous three minutes back on, the total time that has passed when the diver reaches the surface is four minutes and 48 seconds.

Let’s now summarize the key points from this video. If two quantities 𝑥 and 𝑦 are proportional, then they vary at the same rate. And they can be related by the equation 𝑦 equals 𝑘𝑥, where 𝑘 is a constant known as the constant of proportionality. The graph of a proportional relationship is a straight line passing through the origin. If two quantities 𝑥 and 𝑦 obey a linear model, they are related by 𝑦 equals 𝑘𝑥 plus 𝑐, where 𝑘 and 𝑐 are constants.

The graph of any linear model is a straight line. The slope of this line is the value of 𝑘 in the linear model. And the 𝑦-intercept is the value of 𝑐. 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 also saw that we can answer real-world problems approximated by linear models by considering their graph and in particular by examining points that lie on the line, its slope, and its intercepts with the 𝑥- and 𝑦-axes.