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Lines and linear relationships

Lines and linear relations are key to the mathematics of the Cartesian plane. Watch Norman Wildberger explain the fundamentals about lines.
12.4
In this activity, we’re going to be looking at extending our understanding of direct proportions to more general linear relationships. And we’re going to be using the Cartesian framework introduced by Descartes to do that. So we’re looking at general lines. We’ll be able to talk not just about their slopes, but also their intercepts, x and y-intercepts, and really positioning them in the Cartesian plane, getting good control over where they are. We’ll then be able to apply this understanding to important examples– in particular, to temperature measured in the Fahrenheit and Celsius systems and also about the important subject of supply and demand in economics.
53.1
So it all rests on really understanding the geometry of the lines and their equations in the Cartesian setup. So to get at a more general line, let’s have a look at our line l and translate it so that it goes through, say, this particular point here. What would that involve? Well, if we’re wanting a line which is still parallel to this one, it still should have the property that if we go over 2, we go up 1. So if it’s going to go through this point, it should also go through this point here. So that’s over 2 and up 1, and then over 2 up 1, it should go through that point.
91.3
And over 2 up 1, should go through that point. So let’s draw such a line.
99.4
Take our ruler and connect some of our points. And we’ll draw it in purple.
119.2
Good. So this is a new line. Let’s give it a new name. Let’s say Line K. And the natural question is, what would be the equation of this line? What is the relationship between points lying on this line? So let’s have a look. So this point here is the point minus 1, 1. So this point here is the point 1, 2. This is the point 3, 3. This is the point 5, 4. So what’s the relationship between these various x and y-coordinates for this line here? Well, it’s going to be pretty close to the line l that it’s translate of.
170.4
In fact, we’ll be able to write it as y equals 1/2 x plus some adjustment, plus or minus some adjustment. Let’s see what that might be. So if x is equal to minus 1 and y is equal to 1, then we would plug minus 1 in there and y equals 1 there. Then we see in order to make that happen, we would have to put a 3/2 there. Now they would actually be satisfied. Let’s check about this point here– if x is 1, then the right hand side is 1/2 plus 3/2, which is, in fact, 2. So you can check that all of these points here actually satisfy this equation here.
215.4
So this is the equation of the line k, which is a translate of the line l. And it has the same slope. Slope is 1/2 there. That slope is the same. That’s still telling us that if we go over 1, we’re going up 1/2. What’s the significance of this 3/2? Well, the 3/2 is– it’s called the y-intercept.
256.9
That’s the value of y when x is 0. So when x equals 0, then y has to be exactly 3/2. That’s corresponding to this point right here on the y-axis where the line crosses the y-axis. That’s why it’s called the y-intercept. So another alternate form for this line would be to multiply by 2 and then get the x’s and y’s on the same side. So that would be x minus 2y equals minus 3. So you can check that that’s another form for a line equation that describes this purple line. So the general situation that we’re talking about here is we’re talking about lines of the form y equals mx plus b.
309.6
This is a line with slope m and y-intercept b. In other words, when x is 0, then y equals b.
Lines are represented in the Cartesian plane by linear equations, usually in the forms \(\normalsize{ax+by=c}\) or \(\normalsize{y=mx+b}\). The second form emphasizes the importance of the slope \(\normalsize{m}\) and the \(\normalsize{y}\)-intercept \(\normalsize{b}\).
The \(\normalsize{y=mx+b}\) form of a line is very convenient if we want to think of the line as representing a function which inputs a value \(\normalsize{x}\), and outputs another value \(\normalsize{y}\). To emphasize this functional aspect, it is common to also introduce a specific name of the function in question, say \(\normalsize{f}\). Thus we would write
\[\Large{y=f(x)=-2x+4}\]
and sometimes we dispense with the reference to \(\normalsize{y}\), so writing \(\normalsize{f(x)=-2x+4}\).
Picture of the same line, now as a function
For example you can verify that \(\normalsize{f(0)=4}\), \(\normalsize{f(1)=2}\), \(\normalsize{f(10)=-16}\) and \(\normalsize{f(-3)=10}\).
Any function of the form \(\normalsize{f(x)=ax+b}\) for fixed \(\normalsize{a,\;b}\) is called a linear function provided \({\normalsize a \neq 0}\). We say that the physical line on the page for \(\normalsize{y=ax+b}\) is the graph of that function.
The line \(\normalsize x=0\) is not a function. This is because when \(x=0\) there are too many possible values for \(\normalsize y\). In this case we use a more general term, and call \(\normalsize x=0\) a relation.
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Maths for Humans: Linear, Quadratic & Inverse Relations

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