# Forms of Linear Equations

### Slope-Intercept Form The equation of a nonvertical line with slope $m$ and $y$-intercept $b$ can be written in the slope-intercept form $y = mx + b$.
The slope-intercept form of a line is a linear equation, where $m$ is the slope and $b$ is the $y$-intercept of the line, expressed as:
$y=mx+b$
As the name suggests, the slope-intercept form gives two important pieces of information: the slope and the $y$-intercept. This information can be used to graph the line without making a table and plotting points.
Step-By-Step Example
Graphing Positive Slope in Slope-Intercept Form
Graph the line:
$y=\frac{3}{2}x+5$
Step 1
The equation is in slope-intercept form, where $m$ is the slope and $b$ is the $y$-intercept:
$y=mx+b$
Identify the slope and $y$-intercept of the equation.
Step 2

The $y$-intercept is 5, so plot one point at $(0, 5)$.

Then use the rise and run of the slope, $m$, to locate another point.
\begin{aligned}m&=\frac{\text{Rise}}{\text{Run}}\\&=\frac{3}{2}\end{aligned}
The rise is 3, so move up 3 units from the first point $(0, 5)$. The run is 2, so move right 2 units. This means that another point on the line is $(2, 8)$.
Solution
Connect the two points to graph the line.
Step-By-Step Example
Graphing with Negative Slope in Slope-Intercept Form
Graph the line:
$y=-2x-3$
Step 1
The equation is in slope-intercept form, where $m$ is the slope and $b$ is the $y$-intercept. Identify the slope and $y$-intercept of the equation:
\begin{aligned}y&=mx+b\\y&=-2x-3\end{aligned}
The slope is –2. The $y$-intercept is –3.
Step 2

The $y$-intercept is –3, so plot one point at $(0,-3)$.

Then use the rise and run of the slope, $m$, to locate another point. Write the slope as a fraction with a denominator of 1.
\begin{aligned}m&=\frac{\text{Rise}}{\text{Run}}\\&=\frac{-2}{1}\end{aligned}
The rise is –2, so move 2 units down. The run is 1, so move 1 unit to the right. So, another point on the line is $(1,-5)$.
Solution
Connect the two points to graph the line.

### Point-Slope Form The equation of a nonvertical line with slope $m$ that passes through the point $(x_1, y_1)$ can be written in the point-slope form $y-y_1=m(x-x_1)$.
The point-slope form of a line is a linear equation, where $m$ is the slope and $(x_1, y_1)$ is a point on the line, expressed as:
$y-y_1=m(x-x_1)$
As the name suggests, the point-slope form is convenient to use in writing the equation of a line when the slope and the coordinates of a point on the line are given.
Step-By-Step Example
Writing an Equation with a Given Slope and Point
Write the equation of a line with a slope of 3 that passes through the point $(-4,1)$.
Step 1
Substitute 3 for $m$, –4 for $x_1$, and 1 for $y_1$ in point-slope form.
\begin{aligned}y-y_1&=m(x-x_1)\\y-1&=3(x-(-4))\\y-1&=3(x+4)\end{aligned}
Step 2
Although the point-slope form of a line is a correct form, it is common to rewrite the equation of a line in slope-intercept form. First, distribute the 3 on the right side.
\begin{aligned}y-1&=3(x+4)\\y-1&=3x+12\end{aligned}
Solution
\begin{aligned}y-1&=3x+12\\\underline{\phantom{y}+\ 1}&\phantom{=\:}\underline{\phantom{3x\ }+\ \: 1}\\y&=3x+13\end{aligned}
Step-By-Step Example
Graphing a Line in Point-Slope Form
Graph the line of the equation:
$y-4=-3(x+2)$
Step 1

Identify the slope and a point on the line.

The equation uses the point-slope form, where $m$ is the slope and $x_1$ and $y_1$ are coordinates of a point on the line, or $(x_1, y_1)$:
$y-y_1=m(x-x_1)$
When $x_1$ or $y_1$ are being subtracted in the equation, the coordinates are positive.
When $x_1$ or $y_1$ are being added in the equation, the coordinates are negative.
Step 2
Plot the point at $(-2, 4)$. Then use the rise and run of the slope, $m$, to locate another point. Write the slope as:
\begin{aligned}m&=\frac{\text{Rise}}{\text{Run}}\\&=\frac{-3}{1}\end{aligned}
The rise is –3, so move down 3 units. The run is 1, so move right 1 unit.
Solution
Connect the two points to graph the line.
Step-By-Step Example
Determining the Equation of a Line through Two Points
Identify the equation of a line that passes through point $(3, 1)$ and point $(5, 9)$.
Step 1
Use the coordinates of the two points to identify the slope.
\begin{aligned}m &=\frac{y_2-y_1}{x_2-x_1}\\&=\frac{9-1}{5-3}\\&=\frac{8}{2}\\&=4\end{aligned}
Step 2
Choose one of the points on the line to write an equation in point-slope form. Note that using either point will result in correct equations.
$(x_1,y_1)=(3,1)$
Solution
Use the slope and one of the points on the line to write the point-slope form:
$y-1=4(x-3)$
It is common to rewrite the equation in slope-intercept form. First, distribute the 4. Then add 1 to both sides.
\begin{aligned}y-1&=4(x-3)\\y-1&=4x-12\\y-1+1&=4x-12+1\\y&=4x-11\end{aligned}

### Standard Form of a Line The standard form of a line is $Ax + By = C$. It is typically used for determining the $x$- and $y$-intercepts of a line.
The standard form of a line is a linear equation, where $A$, $B$, and $C$ are integers, $A$ is nonnegative, and $A$ and $B$ are both not zero, expressed as:
$Ax+By=C$
Although the standard form of a line is not the most commonly used for graphing, it is called standard because it is used in other applications, such as solving systems of equations. It is also a useful form for finding the intercepts of a line. The same process is used to find the intercepts of a line in any form.
• To calculate the $x$-intercept, set $y=0$ and solve for $x$.
• To calculate the $y$-intercept, set $x=0$ and solve for $y$.
Step-By-Step Example
Identifying the Intercepts of a Line in Standard Form
Identify the $x$-intercept and the $y$-intercept of the equation:
$3x + 4y = 12$
Step 1
Determine the $x$-intercept. Substitute zero for $y$, and solve for $x$.
\begin{aligned}3x + 4y &= 12\\3x+4(0)&=12\\3x&=12\\x&=4\end{aligned}
Step 2
Determine the $y$-intercept. Substitute zero for $x$, and solve for $y$.
\begin{aligned}3x + 4y &= 12\\3(0)+4y&=12\\4y&=12\\y&=3\end{aligned}
Solution
The $x$-intercept is $(4,0)$. The $y$-intercept is $(0,3)$.

Although it is possible to graph a line in standard form, it is sometimes convenient to rewrite the equation in slope-intercept form. One reason is to determine the slope. Another reason is that it is easier to compare two different lines when they are both written in slope-intercept form.

For a line that is given in standard form $Ax+By=C$, to rewrite the equation in slope-intercept form, first subtract $-Ax$ from both sides. Then divide both sides by $B$, and simplify.

Step-By-Step Example
Rewriting the Standard Form of a Line in Slope-Intercept Form
Write the linear equation in slope-intercept form:
$3x + 4y = 12$
Step 1
The equation is in standard form:
\begin{aligned}Ax+By&=C\\3x+4y&=12\end{aligned}
Rewrite the equation to the slope-intercept form:
$y=mx+b$
Subtract $-3x$ from both sides to isolate the $y$-term. Since the slope-intercept form of a line has the $x$-term before the constant, write the $x$-term first on the right side of the equation.
\begin{aligned}3x+4y&=12\\3x+4y-3x&=12-3x\\4y&=-3x+12\end{aligned}
Step 2
Divide both sides by 4.
\begin{aligned}4y&={-3x+12}\\{\frac{4y}{4}}&=\frac{-3x+12}{4}\end{aligned}
Solution
Simplify the equation.
\begin{aligned}\frac{4y}{4}&=\frac{-3x+12}{4}\\y&=-\frac{3}{4}x+3\end{aligned}
The slope is $-\frac{3}{4}$. The $y$-intercept is 3.