In geometry, we have seen the lines drawn on the coordinate plane. To predict whether the lines are parallel, perpendicular or at any angle without using a geometric tool, the best way to find out is by measuring the slope. In this article, we will discuss in detail what a slope is, the slope formula for parallel lines, perpendicular lines, the slope for collinearity with many solved examples.
What is a Slope?
In math, the slope of a line is the change in the y-coordinate relative to the change in the x-coordinate.
The net change in the y-coordinate is represented by Δy and the net change in the x-coordinate is represented by Δx.
Therefore, the change in the y-coordinate in relation to the change in the x-coordinate is given by
m = change in y/change in x = Δy/Δx
Where “m” is the slope of a line.
The slope of the line can also be represented as tan θ = Δy/Δx
therefore tan θ is the slope of a straight line.
In general, the slope of a straight line is a measure of its steepness and direction. The slope of a straight line between two points (x1,y1) and (x2,y2) can be easily determined by finding the difference between the coordinates of the points. The slope is usually represented by the letter “m”.
Slope Formula
If P(x1,y1) and Q(x2,y2) are the two points on a straight line, then the slope formula is as follow:
| Slope, m = Change in y-coordinates/Change in x-coordinatesm m= (y2 – y1)/(x2 – x1) |
Based on the above formula, we can therefore easily calculate the slope of a line between two points.
In other words, the slope of a line between two points is the slope of the line from one point to the other (along the y-axis) over the distance (along the x-axis). Therefore,
Slope, m = Rise/Run
Slope of Line Equation
The equation for the slope of a line and the points, also known as the point-slope form of the equation of a line, is given by:
Whereas the slope-intercept form of the equation of a line.
| y − y1 = m(x − x1) |
Whereas the slope-intercept form the equation of the line is given by:
y = mx + b
Where b is the y-intercept.
How Slope of a line in a graph?
In the given graph, if the angle of inclination of the given line with the x-axis is θ, then the slope of the line is given by tan θ. There is therefore a relationship between lines and angles. In this article, you will learn about various formulas related to angles and lines.
The slope of a line is given as m = tan θ. If two points A (x1, y1) and B (x2, y2) lie on the line with x1 ≠ x2, then the slope of the line AB is as follow : m = tan θ=y2-y1/x2-x1
Where θ is the angle formed by the straight line AB with the positive direction of the x-axis. θ is between 0° and 180°.
It should be noted that θ = 90° is only possible if the line runs parallel to the y-axis, i.e. for x1 = x2, the slope of the line at this particular angle is undefined.
The conditions for the perpendicularity, parallelism and collinearity of straight lines are given below:
Slope for parallel lines
Let us consider two parallel lines given by l1 and l2 with slopes α and β respectively. For two lines to be parallel, their inclination must also be the same, i.e. α=β. This results in the fact that tan α = tan β. The condition that two lines with slopes α, β are parallel is therefore tan α = tan β.
So if the slopes of two lines in the Cartesian plane are equal, then the lines are parallel to each other.
So if two lines are parallel, then m1 = m2.
If you generalize this for n lines, they are only parallel if the slopes of all lines are the same.
If the equation of the two lines ax + by + c = 0 and a’ x + b’ y + c’= 0, then they are parallel if ab’ = a’b. (How? You get this result by determining the slopes of the two lines and equating them)
Slope for perpendicular lines
In the figure, we have two lines l1 and l2 with slopes α, β. If they are perpendicular to each other, we can say that β = α + 90°. (Using the properties of angles)
Their slopes can be expressed as follows:
m1 = tan(α + 90°) and m2 = tan α.
For two lines to be perpendicular to each other, the product of their slope must equal -1.
If the equations of the two lines are given by ax + by + c = 0 and a’ x + b’ y + c’ = 0, then they are perpendicular if aa’+ bb’ = 0. (This result can also be obtained by determining the slopes of the two lines and equating their product to -1)
Slope for collinearity
For two lines AB and BC to be collinear, the slope of both lines must be the same and there must be at least one common point through which they run. For three points A, B and C to be collinear, the slopes of AB and BC must be equal.
If the equation of the two lines is given by ax + by + c = 0 and a’ x+b’ y+c’ = 0, then they are collinear if ab’ c’ = a’ b’ c = a’c’b.
Angle between two lines
If two straight lines intersect at a point, the angle between them can be expressed by their slopes and is given by the this formula
where m1 and m2 are the slopes of the lines AB and CD respectively.
Slope of vertical lines
Vertical lines have no slope as they have no steepness. We can also say that we cannot define the slope of vertical lines.
A vertical line has no values for the x-coordinates. According to the formula for the slope of the line,
Slope, m = (y2 – y1)/(x2 – x1)
However, the following applies to vertical lines: x2 = x1 = 0
From this follows,
m = (y2 – y1)/0 = undefined
In the same way, the slope of the horizontal line is equal to 0, as the y-coordinates are zero.
m = 0/(x2 – x1) = 0 [for the horizontal line]
Positive and Slope
If the value of the slope of a line is positive, this shows that the line rises as it moves upwards or that the slope over the distance is positive.
If the value of the slope is negative, then the line in the diagram goes down when we move along the x-axis.
Frequently Asked Questions Slope of a line
What is the slope of a line?
The slope of a line describes its steepness and direction. It is calculated as the change in y divided by the change in x between any two points on the line. Imagine climbing a certain distance (change in y) and walking a certain distance (change in x). The slope how much you climb per unit run.
what is the slope of a line Formula:
Slope (m) = (y2 – y1) / (x2 – x1)
where (x1, y1) and (x2, y2) are any two points on the line.
What does the slope tell us?
- Positive slope: The line slopes upwards to the right as y increases with increasing x.
- Negative slope: The line slopes downwards to the right, as y decreases as x increases.
- Zero slope: The line is horizontal, no change in y as x changes.
- Indeterminate slope: The line is vertical, infinite change in y with no change in x.
How to determine the slope from the equation of a straight line:
Slope intercept (y = mx + b): the slope (m) is the coefficient of x.
Form of the point slope (y – y1 = m(x – x1)): The slope (m) is the constant multiplier of (x – x1).
General form (ax + by + c = 0): Solve y = mx + b (you may need to rearrange the equation).
What are the Applications of slope?
Rate of change: in real-world scenarios, the slope can represent a speed, a price change, a temperature change, etc.
Parallel and perpendicular lines: Lines with the same slope are parallel, lines with slopes whose product is -1 are perpendicular.
Graphing lines: Understanding slope helps you visualize and sketch lines accurately.
