In Section Distance, we concerned ourselves with the finite line segment between two points \(P\) and \(Q\). Specifically, we found its length (the distance between \(P\) and \(Q\)) and its midpoint. In this section, our focus will be on the entire line, and ways to describe it algebraically. Consider the generic situation below.

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To give a sense of the ‘steepness’ of the line, we recall that we can compute the slope of the line as follows. (Read the character \(\Delta \) as ‘change in’.)

A couple of notes about Equation slope are in order. First, don’t ask why we use the letter ‘\(m\)’ to represent slope. There are many explanations out there, but apparently no one really knows for sure. Secondly, the stipulation \(x_1 \neq x_0\) (or \(\Delta x \neq 0\)) ensures that we aren’t trying to divide by zero. The reader is invited to pause to think about what is happening geometrically when the ‘change in \(x\)’ is \(0\); the anxious reader can skip along to the next example.

A few comments about Example slopeex are in order. First, if the slope is positive then the resulting line is said to be ‘increasing’, meaning as we move from left to right, the \(y\)-values are getting larger. Similarly, if the slope is negative, we say the line is ‘decreasing’, since as we move from left to right, the \(y\)-values are getting smaller. A slope of \(0\) results in a horizontal line which we say is ‘constant’, since the \(y\)-values here remain unchanged as we move from left to right, and an undefined slope results in a vertical line.

Second, the larger the slope is in absolute value, the steeper the line. You may recall from Intermediate Algebra that slope can be described as the ratio ‘\(\frac {\text {rise}}{\text {run}}\)’. For example, if the slope works out to be \(\frac {1}{2}\), we can interpret this as a ‘rise’ of \(1\) unit upward for every ‘run’ of \(2\) units to the right:

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In this way, we may view the slope as ‘the rate of change of \(y\) with respect to \(x\)’. From the expression

\[ m = \frac {\Delta y}{\Delta x}\]

we get \(\Delta y = m \Delta x\) so that the \(y\)-values change ‘\(m\)’ times as fast as the \(x\)-values. We’ll have more to say about this concept in Section LinearFunctions when we explore applications of linear functions; presently, we will keep our attention focused on the analytic geometry of lines. To that end, our next task is to find algebraic equations that describe lines and we start with a discussion of vertical and horizontal lines.

Consider the two lines shown below: \(V\) (for ‘V’ertical Line) and \(H\) (for ‘H’orizontal Line).

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All of the points on the line \(V\) have an \(x\)-coordinate of \(3\). Conversely, any point with an \(x\)-coordinate of \(3\) lies on the line \(V\). Said differently, the point \((x,y)\) lies on \(V\) if and only if \(x = 3\). Because of this, we say the equation \(x=3\) describes the line \(V\), or, said differently, the graph of the equation \(x=3\) is the line \(V\).

In Section Relations, we’ll spend a great deal of time talking about graphing equations. For now, it suffices to know that a graph of an equation is a plot of all of the points which make the equation true. So to graph \(x=3\), we plot all of the points \((x,y)\) which satisfy \(x = 3\) and this gives us our vertical line \(V\).

Turning our attention to \(H\), we note that every point on \(H\) has a \(y\)-coordinate of \(-2\), and vice-versa. Hence the equation \(y = -2\) describes the line \(H\), or the graph of the equation \(y=-2\) is \(H\). In general:

Of course, we may be working on axes which aren’t labeled with the ‘usual’ \(x\)’s and \(y\)’s. In this case, we understand Equation verticalhorizontallines to say ‘\(\text {horizontal axis label} = a\)’ describes a vertical line through \((a,0)\) and ‘\(\text {vertical axis label} = b\)’ describes a horizontal line through \((0,b)\).

Using the concept of slope, we can develop equations for the other varieties of lines. Suppose a line has a slope of \(m\) and contains the point \(\left (x_0, y_0\right )\). Suppose \((x,y)\) is another point on the line, as indicated below.

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Equation slope yields

\[ \begin{array}{rclr} m & = & \frac {y - y_0}{x-x_0} & \\ m\left (x - x_0\right ) & = & y - y_0 & \\ y - y_0 & = & m\left (x - x_0\right ) & \\ \end{array} \]

which is known as the point-slope form of a line.

A few remarks about Equation pointslope are in order. First, note that if the slope \(m = 0\), then the line is horizontal and Equation pointslope reduces to \(y - y_0 = 0\) or \(y = y_0\), as prescribed by Equation verticalhorizontallines. Second, we may need to change the letters in Equation pointslope from ‘\(x\)’ and ‘\(y\)’ depending on the context, so while Equation pointslope should be committed to memory, it should be understood that ‘\(x\)’ refers to whichever variable is used to label the horizontal axis, and \(y\) refers to whichever variable is used to label the vertical axis. Lastly, while Equation pointslope is, by far, the easiest way to construct the equation of a line given a point and a slope, more often than not, the equation is solved for \(y\) and simplified into the form below.

Equation slopeintercept is probably a familiar sight from Intermediate Algebra. You may recall from that class that the ‘intercept’ in ‘slope-intercept’ comes from the fact that this line ‘intercepts’ or crosses the \(y\)-axis at the point \((0,b)\). If we set the slope, \(m = 0\), we obtain \(y=b\), the formula for Horizontal Lines first introduced in Equation verticalhorizontallines. Hence, any line which has a defined slope \(m\) can be represented in both point-slope and slope-intercept forms. The only exceptions are vertical lines. There is one equation - the aptly named ’general form’ - which describes every type of line and it is presented next.

Note the indefinite article ‘a’ in Equation generallineform. The line \(y = 5\) is a general form for the horizontal line through \((0,5)\), but so are \(3y = 15\) and \(0.5 y = 2.5\). The reader is left to ponder the use of the definite article ‘the’ in Equations pointslope and slopeintercept. Regardless of which form the equation of a line takes, note that the variables involved are all raised to the first power. For instance, there are no \(\sqrt {x}\) terms, no \(y^2\) terms or any variables appearing in denominators. Let’s look at a few examples.

While every point on a line holds value and meaning, we’ve reminded you of certain points, called ‘intercepts,’ which hold special enough significance to be singled out. Formally, we define these as follows.

As usual, the labels of the axes in the problem will dictate the labels on the intercepts. If we’re working in the \(vw\)-plane, for instance, there would be \(v\)- and \(w\)-intercepts.

The last little bit of analytic geometry we need to review about lines are the concepts of ‘parallel’ and ‘perpendicular’ lines. Parallel lines do not intersect, and hence, parallel lines necessarily have the same slope. Perpendicular lines intersect at a right (\(90^{\circ }\)) angle. The relationship between these slopes is somewhat more complicated, and is summarized below.

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A few remarks about Theorem parallelperpendicularslopetheorem are in order. First off, the theorem assumes that the slopes of the lines exist. The reader is encouraged to think about the case when one (or both) of the slopes don’t exist. Along those same lines, the reader is encouraged to think about why the stipulations \(m_1 \neq 0\) and \(m_2 \neq 0\) appear in the statement regarding slopes of perpendicular lines, and what happens in this case as well. (Think geometrically!) In Exercise perpendicularlineproof, you’ll prove the assertion about the slopes of perpendicular lines. For now, we accept it as true and use it in the following example.

Our last example with lines sets up a fourth kind of symmetry which will be revisited in Section InverseFunctions.