What is a Line?
- A line can be defined as a straight set of points that extend in opposite directions.
- It has no ends in both directions(infinite).
- It has no thickness.
It is one-dimensional.
What is Slope of a Line?
Definition: The slope of a line indicates the steepness and direction of the line.
In mathematics, the slope of a line is the change in the y-coordinate for a corresponding change in the x-coordinate.
The net change in the y-coordinate is denoted by Δy and the net change in the x-coordinate is denoted by Δx. So, the change in the y-coordinate for a change in the x-coordinate is expressed as:
m =\(\frac{\text{(change in y coordinates)}}{\text{(change in x coordinates)}} \)
\( =\frac{\Delta y}{\Delta x} \)
The letter ‘m’ is commonly used to symbolize the slope.
How to Find the Slope of a Line?
1. Find the change in the y coordinates (the rise), \(\Delta y\) is the change in y:
\(\Delta y=y_2-y_1\)
2. Find the change in the x coordinates (the run), \(\Delta x\) is the change in x:
\(\Delta x=x_2-x_1\)
3. To find the slope, we divide \(\Delta y \) by \(\Delta x\)
\(m=\frac{rise}{run}\)
\(m=\frac{\Delta y}{\Delta x}\)=\(\frac{change~in~y}{change~in~x}\)
\(=\frac{y_2-y_1}{x_2-x_1}\)
Where m is the slope of the given line.
Other Related Topics
(i) The line is in the rising state if the slope is positive, i.e. m > 0.
(ii) When the slope is negative, as in m < 0, the line descends.
(iii) A horizontal line has no slope related to it.
(iv) The slope of a line that is vertical is undefined.
Parallel Lines
Parallel lines are defined as two or more lines that lie in the same plane but never intersect. They have the same slope and are equidistant from each other.
Observe the parallel lines in the figure where a is parallel to b, and p is parallel to q.
Horizontal Lines
A horizontal line is a straight line that runs from left to right, or right to left in the coordinate plane, and is parallel to the x-axis. In other words, a horizontal line is a straight line with an intercept only on the y-axis and not on the x-axis. A horizontal line forms the base for flat shapes.
Vertical Lines
A vertical line is a line parallel to the Y-axis in a coordinate plane. It’s a line that runs from the top to the bottom and from the bottom to the top in a coordinate plane. The x-coordinate for any point along this line will be the same. The points of vertical lines, for the examples shown in the image, are (2,0), (3,0), (4,0), and so on.
Solved Slope of a Line Examples
Example 1: Find the inclination of the line between points A = (1, 0) and B = (5, 2).
Solution:
Given, points A = (1, 0) and B = (5, 2).
According to the slope formula, we know that,
Slope of a line, m = \(\frac{(y_2-y_1)}{(x_2-x_1)}\)
\(m=\frac{(2-0)}{(5-1)}\)
\(=\frac{2}{4}\)
\(=\frac{1}{2}\)
Hence, the slope of required line is \(\frac{1}{2}\)
Example 2:
Find the slope of the line between A (-3, 4) and B (1, -2).
Solution:
If given, A (-3, 4) and B (1, -2) are two points.
According to the slope formula, we know that,
Slope of a line, m = \(\frac{(y_2-y_1)}{(x_2-x_1)}\)
Therefore, the inclination of the line,
\(m = \frac{(- 2- 4)}{1 -(-3)}\)
\( =-\frac{6}{4}\)
\(=-\frac{3}{2}\)
Hence, the slope of required line is – \(\frac{3}{2}\)
Example 3:
Randy was looking at a graph and noticed that the raise was 20 units and the run was 10 units. What would be the slope of a line?
Solution:
Given that,
Raise = 20 units
Run = 10 units
We know that linear inclination is defined as the rate of linear rise.
i.e. Slope, \(m=\frac{Rise}{Run}\)
Therefore, slope \(=\frac{20}{10}= 2\) units.
Therefore, the inclination of the line is 2 units.
We need to find the ratio of the difference between the y-coordinates and the x-coordinates of two points, forming a line. The resulting value is the slope of the line. The slope also shows the “line height” near the y-axis over running on the x-axis.
Types of line slopes:
- Positive slope
Negative slope
Zero slope
- Undefined
Slope indicates the steepness and direction of a line. So in general, the slope exhibits how steep a line is.