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.
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. $\square$
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.
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 $t$-intercept, and describe the practical meaning of each.
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. $\square$
Exercises 1.1
Ex 1.1.1 Find the equation of the line through $(1,1)$ and $(-5, -3)$ in the form $y=mx+b$. (answer)
Ex 1.1.2 Find the equation of the line through $(-1,2)$ with slope $-2$ in the form $y=mx+b$. (answer)
Ex 1.1.3 Find the equation of the line through $(-1,1)$ and $(5, -3)$ in the form $y=mx+b$. (answer)
Ex 1.1.4 Change the equation $y-2x=2$ to the form $y=mx+b$, graph the line, and find the $y$-intercept and $x$-intercept. (answer)
Ex 1.1.5 Change the equation $x+y=6$ to the form $y=mx+b$, graph the line, and find the $y$-intercept and $x$-intercept. (answer)
Ex 1.1.6 Change the equation $x=2y-1$ to the form $y=mx+b$, graph the line, and find the $y$-intercept and $x$-intercept. (answer)
Ex 1.1.7 Change the equation $3=2y$ to the form $y=mx+b$, graph the line, and find the $y$-intercept and $x$-intercept. (answer)
Ex 1.1.8 Change the equation $2x+3y+6=0$ to the form $y=mx+b$, graph the line, and find the $y$-intercept and $x$-intercept. (answer)
Ex 1.1.9 Determine whether the lines $3x+6y=7$ and $2x+4y=5$ are parallel. (answer)
Ex 1.1.10 Suppose a triangle in the $x,y$–plane has vertices $(-1,0)$, $(1,0)$ and $(0,2)$. Find the equations of the three lines that lie along the sides of the triangle in $y=mx+b$ form. (answer)
Ex 1.1.11 Suppose that you are driving to Seattle at constant speed. After you have been traveling for an hour you pass a sign saying it is 130 miles to Seattle, and after driving another 20 minutes you pass a sign saying it is 105 miles to Seattle. Using the horizontal axis for the time $t$ and the vertical axis for the distance $y$ from your starting point, graph and find the equation $y=mt+b$ for your distance from your starting point. How long does the trip to Seattle take? (answer)
Ex 1.1.12 Let $x$ stand for temperature in degrees Celsius (centigrade), and let $y$ stand for temperature in degrees Fahrenheit. A temperature of $0^\circ$C corresponds to $32^\circ$F, and a temperature of $100^\circ$C corresponds to $212^\circ$F. Find the equation of the line that relates temperature Fahrenheit $y$ to temperature Celsius $x$ in the form $y=mx+b$. Graph the line, and find the point at which this line intersects $y=x$. What is the practical meaning of this point? (answer)
Ex 1.1.13 A car rental firm has the following charges for a certain type of car: \$25 per day with 100 free miles included, \$0.15 per mile for more than 100 miles. Suppose you want to rent a car for one day, and you know you'll use it for more than 100 miles. What is the equation relating the cost $y$ to the number of miles $x$ that you drive the car? (answer)
Ex 1.1.14 A photocopy store advertises the following prices: 5¢ per copy for the first 20 copies, 4¢ per copy for the 21st through 100th copy, and 3¢ per copy after the 100th copy. Let $x$ be the number of copies, and let $y$ be the total cost of photocopying. (a) Graph the cost as $x$ goes from 0 to 200 copies. (b) Find the equation in the form $y=mx+b$ that tells you the cost of making $x$ copies when $x$ is more than 100. (answer)
Ex 1.1.15 In the Kingdom of Xyg the tax system works as follows. Someone who earns less than 100 gold coins per month pays no tax. Someone who earns between 100 and 1000 gold coins pays tax equal to $10$% of the amount over 100 gold coins that he or she earns. Someone who earns over 1000 gold coins must hand over to the King all of the money earned over 1000 in addition to the tax on the first 1000. (a) Draw a graph of the tax paid $y$ versus the money earned $x$, and give formulas for $y$ in terms of $x$ in each of the regions $0\le x\le 100$, $100\le x\le 1000$, and $x\ge 1000$. (b) Suppose that the King of Xyg decides to use the second of these line segments (for $100\le x\le 1000$) for $x\le 100$ as well. Explain in practical terms what the King is doing, and what the meaning is of the $y$-intercept. (answer)
Ex 1.1.16 The tax for a single taxpayer is described in the figure 1.1.3. Use this information to graph tax versus taxable income (i.e., $x$ is the amount on Form 1040, line 37, and $y$ is the amount on Form 1040, line 38). Find the slope and $y$-intercept of each line that makes up the polygonal graph, up to $x=97620$. (answer)
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Ex 1.1.17 Market research tells you that if you set the price of an item at \$1.50, you will be able to sell 5000 items; and for every 10 cents you lower the price below \$1.50 you will be able to sell another 1000 items. Let $x$ be the number of items you can sell, and let $P$ be the price of an item. (a) Express $P$ linearly in terms of $x$, in other words, express $P$ in the form $P=mx+b$. (b) Express $x$ linearly in terms of $P$. (answer)
Ex 1.1.18 An instructor gives a 100-point final exam, and decides that a score 90 or above will be a grade of 4.0, a score of 40 or below will be a grade of 0.0, and between 40 and 90 the grading will be linear. Let $x$ be the exam score, and let $y$ be the corresponding grade. Find a formula of the form $y=mx+b$ which applies to scores $x$ between 40 and 90. (answer)