1.2: Lines

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.

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 $3/2$.

Contributors

• David Guichard (Whitman College)

• Integrated by Justin Marshall.