# Linear Equations

### Plotting Points A linear equation can be graphed by using a table to generate a list of ordered pairs. When all the ordered pairs are plotted, they lie on the same line.

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, $y=3x-2$ is a linear equation, but $y=3x^2-2$ 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.

Step-By-Step Example
Using a Table of Ordered Pairs to Graph a Linear Equation
Graph the linear equation:
$y=3x-2$
Step 1

Create a table, and choose a list of $x$-coordinates.

$x$
$y$
$-1$
$0$
$1$
$2$
Step 2

Substitute the chosen coordinates into the equation for $x$, and solve for $y$.

$x$
$y=3x-2$
$y$
$-1$
\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}
$4$
Step 3

Use the table to write a list of ordered pairs.

$x$
$y$
$(x, y)$
$-1$
$-5$
$(-1, -5)$
$0$
$-2$
$(0, -2)$
$1$
$1$
$(1, 1)$
$2$
$4$
$(2, 4)$
Solution
Plot the points, and then connect them with a straight line.

### x- and y-Intercepts The points where a line crosses the $x$- or $y$-axis are called the intercepts. To find the $x$-intercept, let $y = 0$. To find the $y$-intercept, let $x = 0$.

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 x-intercept. 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 y-intercept. 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.

Step-By-Step Example
Using the Intercepts to Graph a Linear Equation
Graph the linear equation:
$y = -5x + 1$
Step 1

Create a table and insert zero for one $x$-coordinate and one $y$-coordinate.

$x$
$y$
$0$
$0$
Step 2

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}
$0$
Step 3

Use the table to write ordered pairs.

$x$
$y$
$(x, y)$
$0$
$1$
$(0, 1)$
$0.2$
$0$
$(0.2, 0)$
Solution
Plot the points, and then connect them with a straight line.

### Slope of a Line The steepness of a line can be described by the slope. The slope of a line is the ratio of the change in $y$-values to the change in $x$-values of any two points on the line.
The slope is the ratio of the change in $y$ to the change in $x$ for two points on a line, represented as:
$\frac{\text{Rise}}{\text{Run}}$
• 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: $y_2-y_1$.
• The run is the horizontal change between two points $(x_1, y_1)$ and $(x_2, y_2)$ on a line: $x_2-x_1$.
• 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:
$m=\frac{\text{Rise}}{\text{Run}}=\frac{y_2-y_1}{x_2-x_1}$
Step-By-Step Example
Determine the Slope of a Line
Identify the slope of the line whose graph is shown.
Step 1

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 $(x_2, y_2)$.

Step 2

Identify the rise and run. There are two ways to identify the rise and run between two points.

Method 1: Subtract the coordinates.
\begin{aligned}\text{Rise}&=y_2-y_1\\&=4-1\\&=3\end{aligned}
\begin{aligned}\text{Run}&=x_2-x_1\\&=2-1\\&=1\end{aligned}
Method 2: Count the squares on the grid.

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.
Solution
Determine the slope, or $m$.
\begin{aligned}m&=\frac{\text{Rise}}{\text{Run}}\\&=\frac{3}{1}\\&=3\end{aligned}
The slope of the line is $m=3$.