Plotting Points
When an equation has two variables, such as $x$ and $y$, the solutions of the equation can be represented by a graph in the coordinate plane. The graph consists of the set of all points $(x,y)$ that make the equation true. A linear equation in two variables is an equation in two variables whose graph is a line. In a linear equation in two variables, each term can be written as a number, such as 2, or a product of a variable and a number, such as $3x$. The variables are not raised to a power, in the denominator of a fraction, or under a radical, such as a square root. For example, $y=3x2$ is a linear equation, but $y=3x^22$ is not.
The most basic method of graphing lines is by plotting points and connecting them. A minimum of two points is required to graph a line, but additional points may be plotted as a check. A table is a good way to organize the values when plotting points to graph a line.
Create a table, and choose a list of $x$coordinates.
$x$ 
$y$ 

$1$ 

$0$ 

$1$ 

$2$ 
Substitute the chosen coordinates into the equation for $x$, and solve for $y$.
$x$ 
$y=3x2$ 
$y$ 

$1$ 
$\begin{aligned}y &= 3(1)2\\y &=5\end{aligned}$ 
$5$ 
$0$ 
$\begin{aligned}y &= 3(0)2\\y &=2\end{aligned}$ 
$2$ 
$1$ 
$\begin{aligned}y &= 3(1)2\\y &=1\end{aligned}$ 
$1$ 
$2$ 
$\begin{aligned}y &= 3(2)2\\y &=4\end{aligned}$ 
$4$ 
Use the table to write a list of ordered pairs.
$x$ 
$y$ 
$(x, y)$ 

$1$ 
$5$ 
$(1, 5)$ 
$0$ 
$2$ 
$(0, 2)$ 
$1$ 
$1$ 
$(1, 1)$ 
$2$ 
$4$ 
$(2, 4)$ 
x and yIntercepts
Another method of graphing a linear equation is to plot the intercepts. Two points determine a line, which means that when two points are plotted, there is only one line that connects them. Any two points can be used to graph a line, but two points that are usually easy to plot are the intercepts. The value of $x$ where a graph touches or crosses the $x$axis is the xintercept. It lies on the $x$axis, so it always has a $y$coordinate of zero. The value of $y$ where a graph touches or crosses the $y$axis is the yintercept. It lies on the $y$axis, so it always has an $x$coordinate of zero.
To graph a linear equation by finding the intercepts, first set $x=0$ and solve for $y$. Then set $y=0$ and solve for $x$. Plot the intercepts, and draw the line connecting them.
Create a table and insert zero for one $x$coordinate and one $y$coordinate.
$x$ 
$y$ 

$0$ 

$0$ 
Substitute zero into the equation for $y$, and solve for $x$. Substitute zero into the equation for $x$, and solve for $y$.
$x$ 
$y = 5x + 1$ 
$y$ 

$0$ 
$\begin{aligned}y &= 5(0)+1\\y &=1\end{aligned}$ 
$1$ 
$0.2$ 
$\begin{aligned}0 &= 5x+1\\1 &=5x\\\frac{1}{5} &= x\\0.2&=x\end{aligned}$ 
$0$ 
Use the table to write ordered pairs.
$x$ 
$y$ 
$(x, y)$ 

$0$ 
$1$ 
$(0, 1)$ 
$0.2$ 
$0$ 
$(0.2, 0)$ 
Slope of a Line
 Slope is a measure of the steepness of a line.
 A line with a positive slope goes up from left to right.
 A line with a negative slope goes down from left to right.
 The rise is the vertical change between two points $(x_1, y_1)$ and $(x_2, y_2)$ on a line: $y_2y_1$.
 The run is the horizontal change between two points $(x_1, y_1)$ and $(x_2, y_2)$ on a line: $x_2x_1$.
 The slope is represented by the variable $m$.
 The formula for the slope of a line through the points $(x_1, y_1)$ and $(x_2, y_2)$ is:
Identify two points on the line. Any two points used will result in the same final answer.
Let $(1, 1)$ be the point $(x_1, y_1)$. Then let $(2, 4)$ be the point $(x_2, y_2)$.
Identify the rise and run. There are two ways to identify the rise and run between two points.
Method 1: Subtract the coordinates.For the rise, count the number of squares vertically from the point $(1,1)$ to the point $(2,4)$. There are 3 squares for the rise.
For the run, count the number of squares horizontally from the point $(1,1)$ to the point $(2,4)$. There is 1 square for the run.