Formula to find the Slope of a Line
When we represent a linear equation in a coordinate plane, then the representation is known as graphing a linear equation. The graph of a linear equation is a straight line. We graph equations by plotting the ordered pairs that satisfy the equation on the coordinate plane and passing a straight line through them.
The slope of a line is the ratio of rise in the y-axis to the run in the x-axis. In simple words, slope is the ratio of the change in y coordinates to the change in x coordinates. In general, slope is denoted by ‘m’.
Consider two points \((x_1,~y_1)\) and \((x_2,~y_2)\) that satisfies the equation \(y~=~mx+b\). The slope of the line will be given by :
\(m=~\frac{\text{rise}}{\text{run}}\)
\(=~\frac{\text{change in y}}{\text{change in x}}\)
\(=~\frac{y_2~-~y_1}{x_2~-~x_1}\)
Slope of Vertical Line
A vertical line is parallel to the y-axis. That is, there will be no change in the x-axis. The slope will then be:
m = \(~\frac{\text{rise}}{\text{run}}\)
= \(~\frac{\text{change in y}}{\text{change in x}}\)
= \(~\frac{y_2~-~y_1}{x_2~-~x_1}\)
= \(~\frac{y_2~-~y_1}{0}\)
We know that division by zero is undefined. Hence the slope of a vertical line can not be defined.
Slope of Horizontal Line
A horizontal line is parallel to the x-axis. So, there will be no change in the y-axis. The slope will be:
m = \(~\frac{\text{rise}}{\text{run}}\)
= \(~\frac{\text{change in y}}{\text{change in x}}\)
= \(~\frac{y_2~-~y_1}{x_2~-~x_1}\)
= 0
The slope of a horizontal line is 0.
Rapid Call
Slope of line:
m = \(~\frac{\text{rise}}{\text{run}}\)
= \(~\frac{\text{change in y}}{\text{change in x}}\)
= \(~\frac{y_2~-~y_1}{x_2~-~x_1}\)
- The slope of a vertical line is undefined
- The slope of a horizontal line is 0.
Solved Examples
Example 1: Calculate the slope of the given line.
Solution :
Let \((x_1,~y_1)=(-3,~4)\) and \((x_2,~y_2)=(3,~-5)\).
m = \(~\frac{y_2~-~y_1}{x_2~-~x_1}\)
= \(~\frac{-~5~-~4}{3-~(-~3)}\)
= \(~\frac{-~9}{6}\)
= \(~\frac{-~3}{2}\)
Example 2: Identify the slope of the given line.
Solution :
This is a horizontal line hence the slope is 0.
Applying the slope formula,
m = \(~\frac{y_2~-~y_1}{x_2~-~x_1}\)
= \(~\frac{3~-~3}{4~-~(-~2)}\)
= 0
Therefore, the slope is 0.
Example 3: Calculate the slope of the given line.
Solution :
The slope of a vertical line is undefined.
Applying the slope formula,
m = \(~\frac{y_2~-~y_1}{x_2~-~x_1}\)
= \(~\frac{3~-~0}{5~-~5}\)
= \(~\frac{3}{0}\)
Division by zero is undefined. So, the slope is undefined.
Example 4: The table shows the distance y(in km) traveled by car in x minutes. The points in the table form a straight line. Find the slope of the line.
x | 3 | 6 | 9 | 12 |
|---|---|---|---|---|
y | 5 | 7 | 9 | 11 |
Solution :
First, each of the ordered pairs is plotted on the coordinate plane. These points are joined by a straight line.
To find the slope, take any two points from the table and apply the slope formula.
Use points \((x_1,~y_1)=(3,~5)\) and \((x_2,~y_2)=(6,~7)\)
m = \(~\frac{y_2~-~y_1}{x_2~-~x_1}\)
= \(~\frac{7~-~5}{6~-~3}\)
= \(~\frac{2}{3}\)
The slope is \(\frac{2}{3}\), which means that the distance traveled by car increases by 2 kilometers every 3 minutes, or the car travels 23 kilometers every minute.
The standard form of line is Ax + By = C.
If the slope of line \(l_1\) and \(l_2\) is \(m_1\) and \(m_2\) respectively and line \(l_1\left | \right |l_2\). Then the value of \(m_1\) and \(m_2\) is same i.e., \(m_1~=~m_2\).
If the slope of line \(l_1\) and \(l_2\) is \(m_1\) and \(m_2\) respectively and line \(l_1~\perp~ l_2\). Then \(m_1\cdot~m_2=-1 \).
The point slope form of a straight line is given by :
\(y~-~y_1~=~m(x~-~x_1) \).
The graph of a linear equation is a straight line.