How to Calculate Slope: Step-by-Step Guide with Examples
The slope or gradient of a line specifies both the direction and steepness of a line. The slope is often represented by the letter ‘m’. The slope is determined by finding the ratio of “vertical change” to the “horizontal change” between any two different points on a line. The steepness or grade of a line is measured by the absolute value of the shape. A slope with a greater absolute value represents a steeper line. The direction of a line either rises, falls, and is horizontal or vertical.
A line that extends from left to right has a positive run and positive rise, and also yielding a positive slope i.e. m > 0
A line that declines from left to right has a negative run and negative fall, and also yielding a negative slope i.e. m < o
Horizontal lines have a zero positive slope, as they have zero rise and a positive run.
The slope is undefined if the line is vertical as the vertical line has zero rise and any amount of run.
Slope Equation
As we know, Tan θ = \[\frac{Height}{Base}\]
And, we know that between any two given points
\[\frac{Height}{Base}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\]
Finally, we get slope equations as:
\[m=Tan\theta =\frac{\Delta y}{\Delta x}\]
Therefore, this becomes our final slope equation at any given point.
Equation of a Straight Line
The general equation of a straight line is represented in the form of y = mx + c, where m is the gradient and coordinates of the y-intercepts are (0,c).
We can determine the equation of a straight line when the gradient and point on the line are given by using the formula: y - b = m ( x - a)
Here, m represents the gradients and (a, b) is on the line.
Equation of a Line Example
Q. Find the Equation of a Line With Gradient 5, Passing Through the Point ( 4,1).
Solution:
Using the formula y - b = m ( x - a), and substituting the values: m = 5, a = 4, and b = 1
We get, y - 1 = 5 ( x - 4)
y - 1 = 5x - 20
y = 5x - 20 + 1
y = 5x - 19
y = 5x - 19
Therefore, the equation of a line with gradient 5, passing through ( 4,1) is y = 5x - 19
How to Find Equation of Line Passing Through Two Given Points?
If a line passes through two points M (x₁, y₁) and N (x₂, y₂) such that x₁ is not equal to x₂ and y₁ is not equal to y₂, the equation of a line can be found using the formula mentioned below:
\[\frac{y-y_{1}}{y_{2}-y_{1}}=\frac{x-x_{1}}{x_{2}-x_{1}}\]
Here, the gradient (m) can be calculated as :
\[m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\]
Line Perpendicular
Perpendicular lines are two lines that meet at right angles (90⁰). The slopes of the two lines are negative reciprocals of each other. This means when one line has a slope of m, a perpendicular line has a slope of -1/m.
When we get multiply slope m by perpendicular slope -1/m., we get the answer -1.
What Does the Slope of a Velocity Time Graph Give?
Velocity is a term that measures both speed and direction of a moving body. A change in velocity is known as acceleration. When the velocity and time are graphed on the y - axis and the x-axis respectively, then the slope of the velocity-time graph gives the acceleration of the object.
The slope is the ratio of change in the y-axis and change in the x-axis.
Therefore, we can determine the slope by using the following formula:
\[m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\]
Here,
m represents the slope
y₂ - y₁ represents the difference in the unit on y-axis.
x₂ - x₁ represents the difference in the unit on x-axis
Solved Example:
1. Find the Equation of a Line that Passes Through the Two Points (2,3) and ( 6,-5)
Solution:
The equation of a line through the point (2,3) and ( 6,-5) can be determined using the formula:
\[\frac{y-y_{1}}{y_{2}-y_{1}}=\frac{x-x_{1}}{x_{2}-x_{1}}\]
As the gradient (m) is not given, we will find the gradient by using the formula:
\[m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\]
Substituting the values x₁ = 2, x₂ = 6, y₁ = 3, and y₂ = -5 in the above formula, we get,
\[m=\frac{-5-3}{6-2}\]
\[m=\frac{-8}{4}\]
m = -2
Using the formula y - y₁ = m (x - x₁), and substituting the values: m = -2 , x₁ = 2 and y₁ = 3
We get, y - 3 = -2 ( x - 2)
y - 3 = -2x + 4
Therefore, the equation of a line passing through the point (-1,2) and ( 2,4) is 2x + y + 1 = 0.
2. Find the Equation of a Line That Passes Through the Point (2,0) and Has a Gradient -2.
Solution:
Using the formula y - b = m ( x - a), and substituting the values: m = -2, a = 2 and b = 0
We get, y - 0 = - 2 (x - 2)
y - 0 = -2x + 4
y = -2x + 4.
FAQs on Slope in Maths: Definition, Formula & Applications
1. What is the slope of a line in coordinate geometry?
In coordinate geometry, the slope of a line is a number that measures its steepness and direction. It is often described as the 'rise over run'. The 'rise' represents the vertical change (change in y-coordinates), and the 'run' represents the horizontal change (change in x-coordinates) between any two distinct points on the line. It is denoted by the letter 'm'.
2. How is the slope of a line calculated if two points, (x₁, y₁) and (x₂, y₂), are given?
When you have two points on a line, (x₁, y₁) and (x₂, y₂), you can calculate the slope (m) using the formula: m = (y₂ - y₁) / (x₂ - x₁). This formula calculates the ratio of the change in the y-coordinates (vertical change) to the change in the x-coordinates (horizontal change).
3. What do positive, negative, zero, and undefined slopes represent graphically?
The value of a slope gives a clear indication of the line's orientation on a graph:
- Positive Slope: The line rises from left to right. As the x-value increases, the y-value also increases.
- Negative Slope: The line falls from left to right. As the x-value increases, the y-value decreases.
- Zero Slope: The line is perfectly horizontal. The y-value remains constant regardless of the x-value.
- Undefined Slope: The line is perfectly vertical. The x-value remains constant, and a vertical line has no 'run'.
4. How can you find the slope of a line from its general equation, ax + by + c = 0?
To find the slope from the general equation ax + by + c = 0, you can rearrange it into the slope-intercept form (y = mx + c). By isolating 'y', the equation becomes by = -ax - c, which simplifies to y = (-a/b)x - (c/b). From this form, it's clear that the slope (m) of the line is -a/b.
5. What is the key difference between a line with a zero slope and a line with an undefined slope?
The key difference lies in their orientation and their 'run'. A line with a zero slope is a horizontal line (like the x-axis), where the 'rise' is zero but the 'run' can be any non-zero value (m = 0/run = 0). In contrast, a line with an undefined slope is a vertical line (like the y-axis), where there is a 'rise' but the 'run' is zero. Since division by zero is undefined, the slope is also considered undefined.
6. How does the slope of a line relate to the angle it makes with the positive x-axis?
The slope of a line is directly related to the angle of inclination. Specifically, the slope (m) is equal to the tangent of the angle (θ) that the line makes with the positive direction of the x-axis. This relationship is expressed by the formula m = tan(θ). This is a fundamental concept in trigonometry and coordinate geometry, allowing us to find one value if we know the other.
7. How can we use slope to determine if two lines are parallel or perpendicular?
The slopes of two lines can quickly tell us about their geometric relationship.
- Parallel Lines: Two non-vertical lines are parallel if and only if their slopes are equal (m₁ = m₂).
- Perpendicular Lines: Two non-vertical lines are perpendicular if and only if the product of their slopes is -1 (m₁ * m₂ = -1). This means one slope is the negative reciprocal of the other.
8. Can you give a real-world example where the concept of slope is used?
A common real-world example of slope is in civil engineering and construction. When designing a road, ramp, or roof, engineers must calculate the slope (often called the 'grade' or 'pitch'). For example, a wheelchair ramp must have a very gentle slope to be accessible, while a drainage pipe needs a specific negative slope to ensure water flows away from a building. This ensures safety, functionality, and compliance with building codes.
