Section 1.2 Lines
Subsection 1.2.1 Equations of Lines
When I defined loci in Section 1.1, I mentioned that the first and most common type of locus is a line. This section reviews some definitions and calculations for lines. This should be familiar material from your previous mathematical experience, but it is worth reviewing.
A line is the locus of any equation where the variables \(x\) and \(y\) show up multiplied by constants and added together. Other than multiplication by a constant, no other operations are performed on the variables: no exponents, no roots, no functions, nothing else. These equations, since they produce lines, are called linear equations. Here are some examples.
\begin{align*}
y \amp = 7x - 9 \\
5y + 4x - 10 \amp = 0\\
\frac{1}{5} y - \frac{2}{7} x - \frac{5}{17} \amp = 0 \\
\sqrt{7}y \amp = \pi x + \frac{1}{\sqrt{19}}
\end{align*}
In all these equations, all I have done is multiplied each variable by some constant and added the results together, making them equal to some other constant. By adding and subtracting terms, I can put some of the variables or the constant on the right or left side of the equation, as I see fit.
If I wanted a general form, then I let the symbols \(a_1\text{,}\) \(a_2\) and\(a_3\) stand for constants. I could bring everything over to one side of the equation and write the general form of an equation of a line as follows.
\begin{equation*}
a_1 x + a_2 y + a_3 = 0
\end{equation*}
This is a reasonable general form. All lines can be described by an equation of this form. However, this is not the most intuitive form — I get that form by considering slope.
Subsection 1.2.2 Slope
The slope of a straight line is a measure of its steepness. A large positive slope descrbes a steep line — one that is quickly increasing. A small positive slope describes a shallow line — one that is slowly increasing. A negative slope describes a decreasing line.
A line with a slope of zero is a horizontal line: it does not increase or decrease at all. A vertical line has no slope at all. (It does not have an infinite slope — the slope is simply undefined).
To calculate the slope, I can choose any two points on the line. I calculate the difference in the \(y\) coordinates and call this the rise: it measures the vertical distance between the two points. Then I calculate the difference in the \(x\) coordinates and call this the run: it measures the horizontal distance between the two points. The slope is the rise divided by the run.
\begin{equation*}
\text{Slope} = \frac{\text{Rise}}{\text{Run}}
\end{equation*}
An example of a slope calculation is shown in Figure 1.2.1.
The calculation of slope as rise over run shows again why veritcal lines have no slope. The \(x\) coordinates of any two points on a vertical line are the same, to the run would be zero. To calculate the slope would involve division by zero, which is not defined. Therefore, vertical lines have no slope. (Dividing by zero is undefined — not infinity.)
Subsection 1.2.3 Slope Intercept Form
Now that I have defined slope, I can review or introduce a common way of writing the equations of lines. I can return to the four example lines I use above; in each of these, I will re-arrange the equation so that \(y\) is alone on the left side.
\begin{align*}
y \amp = 7x - 9 \\
y \amp = - \frac{4}{5} x + 2\\
y \amp = \frac{10}{7} x + \frac{25}{17} \\
y \amp = \frac{\pi}{\sqrt{7}} x + \frac{1}{\sqrt{19}\sqrt{7}}
\end{align*}
To get the general form, I solve for \(y\) on the left. Let \(m\) and \(b\) be constants (the choice of these letters is conventional and used in many places). This process produces the well-known slope-intercept form of the equation of a line.
\begin{equation*}
y = mx + b
\end{equation*}
In this form, the number \(m\) will always be the slope of the line. The number \(b\) will be the \(y\)-coordinate of the point where the line crosses the \(y\) axis; this is called the \(y\)-intercept. These two pieces of information are enough to completely determine the line. If I know where a line crosses the \(y\)-axis and I know how quickly is grows from that point (its slope), then I know everything about the line. Figure 1.2.2 shows the general idea of this slope-intercept form.
As I said above, a horizontal line has a slope of zero. In slope-intercept form this is \(m = 0\text{,}\) so the equation of the line looks like
\begin{equation*}
y = 0x + b = b \text{.}
\end{equation*}
Horizontal lines have equations \(y = b\) for some constant. All I need to specify is the \(y\) coordinate. \(y = 4\) is the horizontal line where the \(y\)-coordinate is always \(4\text{,}\) and the \(x\)-coordinate can be anything whatsoever. Figure 1.2.3 shows some horizontal lines.
I’ve said several times that a vertical line has no slope; therefore, it cannot be expressed in slope-intercept form. However, the previous paragraph gives me a good idea of how to understand vertical lines. If a horizontal line is simply determined by its \(y\) coordinate, a vertical line is similarly determined by its \(x\) coordinate. Any line of the form \(x = b\) is a vertical line. \(x = -3\) is the vertical line of points with \(x\)-coordinate \(-3\) and any \(y\) coordinate whatsoever. Figure 1.2.4 shows some vertical lines.
Subsection 1.2.4 Calcluating Equations of Lines
It is use to know how to calculate the equation of a line given various pieces of information about this line.
- If I am given the slope and the \(y\)-intercept, I just put those numbers directly into the slope-intercept form. For example, if I was asked for the line with slope \(\frac{-2}{5}\) and \(y\)-intercept \(3\text{,}\) I would simply put those numbers into the form and get the equation of the line.\begin{equation*} y = \frac{-2}{5} x + 3 \text{.} \end{equation*}
- Sometimes I am given a point and a slope. For example, say I was asked for the equation of the line with slope \(4\) that goes through the point \((2,2)\text{.}\) I have the slope, so I can put that in place of \(m\) in the slope intercept form.\begin{equation*} y = 4x + b \end{equation*}The \(y\)-intercept is still unknown. However, if I substitute the point, I can solve for \(b\text{.}\)\begin{align*} y \amp = 4x + b \\ 2 \amp = (4)(2) + b \\ 2 \amp = 8 + b \\ -6 \amp = b \end{align*}That gives me a value for the \(y\)-intercept, hence the equation of the line.\begin{equation*} y = 4x - 6 \end{equation*}This process works for any line given a slope and a point: put the slope into the slope-intercept form, put the point in for the coordinates \(x\) and \(y\text{,}\) and solve for the intercept.
- There is one last way of describing a line: by giving two points. There is a unique line passing between any two points — but how do I get the equation of this line? I calculate the slope. Recall that slope is defined as the quotient of rise over run. Rise is the difference in the \(y\) coordinates and run is the difference in the \(x\) coordinates, so given two points I can calculate the slope. I’ll show this by example: what is the line through the point \((3,-5)\) and \((1,1)\text{?}\) First I’ll calculate the slope.\begin{equation*} m = \frac{\text{Rise}}{\text{Run}} = \frac{-5 - 1}{3 - 1} = \frac{-6}{2} = -3 \end{equation*}The slope is \(-3\text{.}\) I can write the slope-intercept form without a still-unkown intercept.\begin{equation*} y = -3x + b \end{equation*}Then the process is the same as the previous case: I have the slope and I have a point (two, in fact). I can substitue a point (either point will work) to calculate the intercept.\begin{align*} y \amp = -3x + b \\ -5 \amp = (-3)(3) + b \\ -5 \amp = -9 + b \\ 4 \amp = b \end{align*}With the intercept, I can write the equation of the line.\begin{equation*} y = -3x + 4 \end{equation*}I’ll leave it up to you to check if I had used the other point, I would have calculated the same value for \(b\text{.}\)
