1.2: Lines

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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 $(5+2)/(3-1)=7/2$.

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

$alt$

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 $3/2$.

. 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.

Contributors

David Guichard (Whitman College)
Integrated by Justin Marshall.