Section 5.1
Writing Equations of Lines
When we deal with equations of lines in a coordinate system, a critical number is the slope, usually denoted by the letter $m$. The slope of a line that passes through the two points $A(x_1,y_1)$ and $( {{x}_{2}},{{y}_{2}})$ has its slope computed as
$$\text{slope}=m =\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}.$$
If ${{y}_{2}}={{y}_{1}}$, then the slope of the line is 0. In this case, the two points $A$ and $B$ have the same y-coordinate. The line is horizontal.
If ${{x}_{2}}={{x}_{1}}$ then the slope of the line is undefined. In this case, the two points $A$ and $B$ have the same x-coordinate, so the line is vertical.
There are three common forms in which equations of lines appear.
Standard Form: $Ax+By=C$, with $A,\text{ }B,\text{ and }C$ constants (and $A\text{ and }B$ not both 0). In graphing a line whose equation appears in this form, let $x=0$ and compute $y$; then let $y=0$ and compute $x$. That gives two points on the line, so the line is easy to graph.
Slope-Intercept Form: $y=mx+b$, with $m$ and $b$ constants.
A line whose equation appears in this form can be graphed easily because when $x=0$, then $y=b$ (this is the $y$-intercept), and $m$ is the slope.
Point-Slope Form: $y-y_1=m( x-x_1)$, with $m,\;x_1,\text{ and } y_1$ all constant.
A line whose equation is written in this form passes through the point $( x_1,y_1)$ and has a slope of $m$.
Slope of a line 2: Slope of a Line 2
Examples
a. The slope of the line passing through the points $( 5,7)$ and $( 2,9)$ is $m=\dfrac{9-7}{2-5}=-\dfrac{2}{3}$.
b. The line $3x-2y=6$ passes through the points $( 0,-3)$ and $( 2,0)$.
c. The line $y=\dfrac{3}{2}x-4$ has a $y-$intercept of $-4$ and slope $m=\dfrac{3}{2}$.
d. The line $y-4=7(x+2)$ passes through the point $( -2,4)$ and has a slope $m=7$.
e. The line $x=-2$ is a vertical line. Every point on the line has an x-coordinate of $-2$.