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Identify the Slope and Intercept of a Line in Slope-Intercept Form
> ... Math > Slopes, Intercepts, Equation of a Straight Line > Identify the Slope and Intercept of a Line in Slope-Intercept Form

Introduction

In this lesson, you will identify the slope and the y-intercept from an equation written in slope-intercept form.

This video illustrates the lesson material below. Watching the video is optional.

Identify the Slope and Intercept

When an equation is in the slope-intercept form, the slope and y-intercept can be identified. Slope is represented by \(m\) and the y-intercept is represented by \(b\).

\begin{align*}\color{black}\large\text{Slope Intercept Form: y=\(m\)x+\(b\)}\\\color{black}\text{where \(m\) = slope, and \(b\) = y-intercept}\\\end{align*}

Example 1
Identify the slope and y-intercept of \(y = 2x + 4\).

\begin{align*}m &= 2 &\color{navy}\small\text{slope}\\b &= 4 &\color{navy}\small\text{y-intercept}\\\end{align*}

The slope is 2 and the y-intercept is 4.

Example 2
Identify the slope and y-intercept of \(y = 3x -2\).

\begin{align*}m &= 3 &\color{navy}\small\text{slope}\\b &= -2 &\color{navy}\small\text{y-intercept}\\\end{align*}

The slope is 3 and the y-intercept is -2.

Example 3
Identify the slope and y-intercept of \(y=-1x+3\).

\begin{align*}m &= -1 &\color{navy}\small\text{slope}\\b &= 3 &\color{navy}\small\text{y-intercept}\\\end{align*}

The slope is -1 and the y-intercept is 3.

The y-intercept is in the upper, or positive, region of the y-axis. The slope is negative, which means that when you move left to right along the line, it goes in a downward direction.

The figure shows that the y-intercept is in the upper, or positive, region of the y-axis. The slope is negative, which means that the line goes from the upper left-hand corner down to the bottom right-hand corner.

Figure 1

Example 4
Identify the slope and y-intercept of \(y=\frac{1}{3}x+0\).

\begin{align*}m &= \frac{1}{3} &\color{navy}\small\text{slope}\\\\b &= 0 &\color{navy}\small\text{y-intercept}\\\end{align*}

The slope is \(\frac{1}{3}\) and the y-intercept is 0.

This equation would usually be written as \(y=\frac{1}{3}x\). If there isn’t a value for b, then b is 0, which means that the line goes through the origin \((0,0)\). The slope is positive, but the run on the bottom of the fraction (3) is larger than the rise on the top of the fraction (1). This means that the slope goes up one unit and runs over three units to the right; it has a flatter slope.

The equation is y=1/3x, indicating a line passing through the origin with a positive slope where the rise is one unit for every three units of horizontal run, resulting in a flatter slope.

Figure 2

Do not worry about the exactness of the coordinates but focus on understanding the meaning of the slope and the y-intercept of an equation.

Things to Remember


  • In slope-intercept form, \(y=mx+b\), \(m\) is the slope and \(b\) is the y-intercept.
  • A negative slope means that when you move left to right along the line, the line goes in a downward direction.
  • A positive slope means that when you move left to right along the line, the line goes in an upward direction.

Practice Problems

1. Find the slope of the line:
\(\text{y}=6\text{x}+2\) (
Solution
x
Solution: 6
Details:
The equation of the line is written in the slope-intercept form, which is: \(y = {\color{green}m}x + b\), where \({\color{green}m}\) represents the \({\color{navy}slope}\) and b represents the y-intercept. In this equation, \(y = {\color{green}6}x + 2\), you see that the slope of the line is \({\color{green}6}\).
)2. Find the y-intercept of the line:
\({\text{y}}=-7{\text{x}}+4\) (
Solution
x
Solution: 4
Details:
The equation of the line is written in the slope-intercept form, which is: \(y = mx + {\color{green}b}\), where m represents the slope and \({\color{green}b}\) represents the \({\color{green}y-intercept}\). In this equation, \(y=-7x+{\color{green}4}\), you see that the \({\color{green}y-intercept}\) of the line is \({\color{green}4}\).
)3. Find the slope of the line:
\({\text{y}}=-3{\text{x}}+5\) (
Solution
x
Solution: \(-3\)
Details:
The equation of the line is written in the slope-intercept form, which is: \(y = {\color{green}m}x + b\), where \({\color{green}m}\) represents the \({\color{navy}slope}\) and b represents the y-intercept. In this equation, \(y={\color{green}-3}x+5\), you see that the slope of the line is \({\color{green}-3}\).
)4. Find the y-intercept of the line:
\({\text{y}}=-{\text{x}}-3\) (
Solution
x
Solution: \(-3\)
)
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