# 1.1: Lines

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

If we have two points \(A(x_1,y_1)\) and \(B(x_2,y_2)\), then we can draw one and only one line through both points. By the *slope* of this line we mean the ratio of \(\Delta y\) to \(\Delta x\). The slope is often denoted \(m\): \(m=\Delta y/\Delta x=(y_2-y_1)/(x_2-x_1)\). For example, the line joining the points \((1,-2)\) and \((3,5)\) has slope \((5+2)/(3-1)=7/2\).

Example 1.1.1

According to the 1990 U.S. federal income tax schedules, a head of household paid 15% on taxable income up to $26050. If taxable income was between $26050 and $134930, then, in addition, 28% was to be paid on the amount between $26050 and $67200, and 33% paid on the amount over $67200 (if any). Interpret the tax bracket information (15%, 28%, or 33%) using mathematical terminology, and graph the tax on the \(y\)-axis against the taxable income on the \(x\)-axis.

**Solution**

The percentages, when converted to decimal values 0.15, 0.28, and 0.33, are the *slopes* of the straight lines which form the graph of the tax for the corresponding tax brackets. The tax graph is what's called a*polygonal** line*, i.e., it's made up of several straight line segments of different slopes. The first line starts at the point (0,0) and heads upward with slope 0.15 (i.e., it goes upward 15 for every increase of 100 in the \(x\)-direction), until it reaches the point above \(x=26050\). Then the graph "bends upward,'' i.e., the slope changes to 0.28. As the horizontal coordinate goes from \(x=26050\) to \(x=67200\), the line goes upward 28 for each 100 in the \(x\)-direction. At \(x=67200\) the line turns upward again and continues with slope 0.33. See figure __1.1.1__.

*Figure 1.1.1. Tax vs. income.*

The most familiar form of the equation of a straight line is: \(y=mx+b\). Here \(m\) is the slope of the line: if you increase \(x\) by 1, the equation tells you that you have to increase \(y\) by \(m\). If you increase \(x\) by \(\Delta x\), then \(y\) increases by \(\Delta y=m\Delta x\). The number \(b\) is called the **y-intercept**, because it is where the line crosses the \(y\)-axis. If you know two points on a line, the formula \(m=(y_2-y_1)/ (x_2-x_1)\) gives you the slope. Once you know a point and the slope, then the \(y\)-intercept can be found by substituting the coordinates of either point in the equation: \(y_1=mx_1+b\), i.e., \(b=y_1-mx_1\). Alternatively, one can use the "point-slope'' form of the equation of a straight line: start with \((y-y_1)/(x-x_1)=m\) and then multiply to get \((y-y_1)=m(x-x_1)\), the point-slope form. Of course, this may be further manipulated to get \(y=mx-mx_1+y_1\), which is essentially the "\(mx+b\)'' form.

It is possible to find the equation of a line between two points directly from the relation \((y-y_1)/(x-x_1)=(y_2-y_1)/(x_2-x_1)\), which says "the slope measured between the point \((x_1,y_1)\) and the point \((x_2,y_2)\) is the same as the slope measured between the point \((x_1,y_1)\) and any other point \((x,y)\) on the line.'' For example, if we want to find the equation of the line joining our earlier points \(A(2,1)\) and \(B(3,3)\), we can use this formula: $$ {y-1\over x-2}={3-1\over 3-2}=2,\qquad\hbox{so that}\qquad y-1=2(x-2),\qquad\hbox{i.e.,}\qquad y=2x-3. $$ Of course, this is really just the point-slope formula, except that we are not computing \(m\) in a separate step.

The slope \(m\) of a line in the form \(y=mx+b\) tells us the direction in which the line is pointing. If \(m\) is positive, the line goes into the 1st quadrant as you go from left to right. If \(m\) is large and positive, it has a steep incline, while if \(m\) is small and positive, then the line has a small angle of inclination. If \(m\) is negative, the line goes into the 4th quadrant as you go from left to right. If \(m\) is a large negative number (large in absolute value), then the line points steeply downward; while if \(m\) is negative but near zero, then it points only a little downward. These four possibilities are illustrated in Figure __1.1.2__

**Figure 1.1.2.** Lines with slopes 3, 0.1, -4, and -0.1

If \(m=0\), then the line is horizontal: its equation is simply \(y=b\).

There is one type of line that cannot be written in the form \(y=mx+b\), namely, vertical lines. A vertical line has an equation of the form \(x=a\). Sometimes one says that a vertical line has an "infinite'' slope.

Sometimes it is useful to find the \(x\)-intercept of a line \(y=mx+b\). This is the \(x\)-value when \(y=0\). Setting \(mx+b\) equal to 0 and solving for \(x\) gives: \(x=-b/m\). For example, the line \(y=2x-3\) through the points \(A(2,1)\) and \(B(3,3)\) has \(x\)-intercept \(3/2\).

Example 1.1.2

Suppose that you are driving to Seattle at constant speed, and notice that after you have been traveling for 1 hour (i.e., \(t=1\)), you pass a sign saying it is 110 miles to Seattle, and after driving another half-hour you pass a sign saying it is 85 miles to Seattle. Using the horizontal axis for the time \(t\) and the vertical axis for the distance \(y\) from Seattle, graph and find the equation \(y=mt+b\) for your distance from Seattle. Find the slope, \(y\)-intercept, and \(t\)-intrcept, and describe the practical meaning of each.

**Solution**

The graph of \(y\) versus \(t\) is a straight line because you are traveling at constant speed. The line passes through the two points \((1,110)\) and \((1.5,85)\), so its slope is \(m=(85-110)/(1.5-1)=-50\). The meaning of the slope is that you are traveling at 50 mph; \(m\) is negative because you are traveling *toward* Seattle, i.e., your distance \(y\) is *decreasing*. The word "velocity'' is often used for \(m=-50\), when we want to indicate direction, while the word "speed'' refers to the magnitude (absolute value) of velocity, which is 50 mph. To find the equation of the line, we use the point-slope formula: $$ {y-110\over t-1}=-50,\qquad\hbox{so that}\qquad y=-50(t-1)+110=-50t+160. $$ The meaning of the \(y\)-intercept 160 is that when \(t=0\) (when you started the trip) you were 160 miles from Seattle. To find the \(t\)-intercept, set \(0=-50t+160\), so that \(t=160/50=3.2\). The meaning of the \(t\)-intercept is the duration of your trip, from the start until you arrive in Seattle. After traveling 3 hours and 12 minutes, your distance \(y\) from Seattle will be 0.