Theory:

In the previous topic, we learnt about the equations and how to plot these equations in graph. Now, let us learn about the new concept slope of the line.
The measure of the steepness of the line is called as the slope of the line.
The ratio of change in \(y\) to the ratio of change in \(x\) is called a slope of a line. The slope of the line is also known as the gradient of the line.
 
The slope of the line is generally denoted by \(m\).
 
The slope of the line is defined as:
 
\(\text{Slope} (m)\) \(=\) \(\frac{\text{rise}}{\text{run}}\) \(=\) \(\frac{\text{vertical change}}{\text{horizontal change}}\)
 
\(=\)\(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
 
Where \((x_{1},y_{1})\),\((x_{2},y_{2})\) are any two coordinates of the line.
Example:
From the given graph, let us find the slope of the line.
 
6.PNG
 
Solution:
 
The slope of the line can be determined using the formula:
 
\(\text{Slope} (m)\) \(=\) \(\frac{rise}{run}\)\(=\)\(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
 
Now, substituting \(x_{1}=2\), \(y_{1}=1\), \(x_{2}=6\) and \(y_{2}=4\) in the above equation, we have:
 
\(= \frac{4-1}{6-2}\)
 
\(= \frac{3}{4}\)
 
Therefore, the slope of the given line is \(\frac{3}{4}\).
The slope is said to be undefined if the denominator of the fraction rise over run is zero.
Example:
From the given graph, let us find the slope.
 
6.PNG
 
Solution:
 
The slope of the line can be determined using the formula:
 
\(\text{Slope} (m)\) \(=\) \(\frac{rise}{run}\)\(=\)\(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
 
Now, substituting \(x_{1}=-2\), \(y_{1}=1\), \(x_{2}=-2\) and \(y_{2}=-3\) in the above equation, we have:
 
\(= \frac{-3-1}{-2-(-2)}\)
 
\(= \frac{-4}{-2+2}\)
 
\(= \) Undefined
 
Therefore, the slope of the given line is undefined.