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Section 3.2 : Lines

Let’s start this section off with a quick mathematical definition of a line. Any equation that can be written in the form,

\[Ax + By = C\]

where we can’t have both \(A\) and \(B\) be zero simultaneously is a line. It is okay if one of them is zero, we just can’t have both be zero. Note that this is sometimes called the standard form of the line.

Before we get too far into this section it would probably be helpful to recall that a line is defined by any two points that are on the line. Given two points that are on the line we can graph the line and/or write down the equation of the line. This fact will be used several times throughout this section.

One of the more important ideas that we’ll be discussing in this section is that of slope. The slope of a line is a measure of the steepness of a line and it can also be used to measure whether a line is increasing or decreasing as we move from left to right. Here is the precise definition of the slope of a line.

Given any two points on the line say, \(\left( {{x_1},{y_1}} \right)\) and \(\left( {{x_2},{y_2}} \right)\), the slope of the line is given by,

\[m = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\]

In other words, the slope is the difference in the \(y\) values divided by the difference in the \(x\) values. Also, do not get worried about the subscripts on the variables. These are used fairly regularly from this point on and are simply used to denote the fact that the variables are both \(x\) or \(y\) values but are, in all likelihood, different.

When using this definition do not worry about which point should be the first point and which point should be the second point. You can choose either to be the first and/or second and we’ll get exactly the same value for the slope.

There is also a geometric “definition” of the slope of the line as well. You will often hear the slope as being defined as follows,

\[m = \frac{{{\mbox{rise}}}}{{{\mbox{run}}}}\]

The two definitions are identical as the following diagram illustrates. The numerators and denominators of both definitions are the same.

Note as well that if we have the slope (written as a fraction) and a point on the line, say \(\left( {{x_1},{y_1}} \right)\), then we can easily find a second point that is also on the line. Before seeing how this can be done let’s take the convention that if the slope is negative we will put the minus sign on the numerator of the slope. In other words, we will assume that the rise is negative if the slope is negative. Note as well that a negative rise is really a fall.

So, we have the slope, written as a fraction, and a point on the line, \(\left( {{x_1},{y_1}} \right)\). To get the coordinates of the second point, \(\left( {{x_2},{y_2}} \right)\) all that we need to do is start at \(\left( {{x_1},{y_1}} \right)\) then move to the right by the run (or denominator of the slope) and then up/down by rise (or the numerator of the slope) depending on the sign of the rise. We can also write down some equations for the coordinates of the second point as follows,

\[\begin{align*}{x_2} & = {x_1} + \,\,{\mbox{run}}\\ {y_2} & = {y_1} + \,\,{\mbox{rise}}\end{align*}\]

Note that if the slope is negative then the rise will be a negative number.

Let’s compute a couple of slopes.

We can use this set of examples to see some general facts about lines.

First, we can see from the first two parts that the larger the number (ignoring any minus signs) the steeper the line. So, we can use the slope to tell us something about just how steep a line is.

Next, we can see that if the slope is a positive number then the line will be increasing as we move from left to right. Likewise, if the slope is a negative number then the line will decrease as we move from left to right.

We can use the final two parts to see what the slopes of horizontal and vertical lines will be. A horizontal line will always have a slope of zero and a vertical line will always have an undefined slope.

We now need to take a look at some special forms of the equation of the line.

We will start off with horizontal and vertical lines. A horizontal line with a \(y\)-intercept of \(b\) will have the equation,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{y = b}}\]

Likewise, a vertical line with an \(x\)-intercept of \(a\) will have the equation,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{x = a}}\]

So, if we go back and look that the last two parts of the previous example we can see that the equation of the line for the horizontal line in the third part is

\[y = 2\]

while the equation for the vertical line in the fourth part is

\[x = 4\]

The next special form of the line that we need to look at is the point-slope form of the line. This form is very useful for writing down the equation of a line. If we know that a line passes through the point \(\left( {{x_1},{y_1}} \right)\) and has a slope of \(m\) then the point-slope form of the equation of the line is,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{y - {y_1} = m\left( {x - {x_1}} \right)}}\]

Sometimes this is written as,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{y = {y_1} + m\left( {x - {x_1}} \right)}}\]

The form it’s written in usually depends on the instructor that is teaching the class.

As stated earlier this form is particularly useful for writing down the equation of a line so let’s take a look at an example of this.

The final special form of the equation of the line is probably the one that most people are familiar with. It is the slope-intercept form. In this case if we know that a line has slope \(m\) and has a \(y\)-intercept of \(\left( {0,b} \right)\) then the slope-intercept form of the equation of the line is,

\[y = mx + b\]

This form is particularly useful for graphing lines. Let’s take a look at a couple of examples.

The final topic that we need to discuss in this section is that of parallel and perpendicular lines. Here is a sketch of parallel and perpendicular lines.

Suppose that the slope of Line 1 is \({m_1}\) and the slope of Line 2 is \({m_2}\). We can relate the slopes of parallel lines and we can relate slopes of perpendicular lines as follows.

\[\begin{align*} & {\mbox{parallel : }} {m_1} = {m_2}\\ & {\mbox{perpendicular : }}{m_1}{m_2} = - 1\,\,\,{\mbox{or }}{m_2} = - \frac{1}{{{m_1}}}\end{align*}\]

Note that there are two forms of the equation for perpendicular lines. The second is the more common and in this case we usually say that \({m_2}\) is the negative reciprocal of \({m_1}\).