How to Find Slope Using Formula Two Points Graph and Equation Methods
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 of a Line Definition Formula and Applications
1. What is slope in mathematics?
The slope of a line is a measure of its steepness and direction, showing how much y changes for every change in x. In coordinate geometry, slope tells us how a line rises or falls on a graph.
- If slope is positive, the line rises from left to right.
- If slope is negative, the line falls from left to right.
- If slope is zero, the line is horizontal.
- If slope is undefined, the line is vertical.
2. What is the formula for slope?
The formula for slope is m = (y₂ − y₁) / (x₂ − x₁). This formula calculates the rate of change between two points on a line.
- (x₁, y₁) and (x₂, y₂) are two points on the line.
- Subtract the y-values: y₂ − y₁.
- Subtract the x-values: x₂ − x₁.
- Divide the change in y by the change in x.
3. How do you calculate the slope between two points?
To calculate slope between two points, use m = (y₂ − y₁) / (x₂ − x₁) and substitute the coordinates. For example, for points (2, 3) and (6, 11):
- Change in y = 11 − 3 = 8
- Change in x = 6 − 2 = 4
- Slope m = 8 / 4 = 2
4. What does a positive or negative slope mean?
A positive slope means the line rises from left to right, while a negative slope means it falls from left to right. In graphs and linear equations:
- Positive slope (m > 0): y increases as x increases.
- Negative slope (m < 0): y decreases as x increases.
- Zero slope: horizontal line.
- Undefined slope: vertical line.
5. What is the slope-intercept form of a line?
The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept. This form makes it easy to identify:
- m: the rate of change (slope).
- b: the point where the line crosses the y-axis.
6. How do you find slope from a graph?
To find slope from a graph, calculate rise over run between two clear points on the line. Follow these steps:
- Pick two points where the line crosses grid intersections.
- Count the vertical change (rise).
- Count the horizontal change (run).
- Divide rise by run: m = rise/run.
7. What is the slope of a horizontal and vertical line?
A horizontal line has a slope of 0, and a vertical line has an undefined slope.
- Horizontal line: y-value stays constant, so change in y = 0.
- Vertical line: x-value stays constant, so change in x = 0, and division by zero is undefined.
8. What is the difference between slope and y-intercept?
The slope measures the rate of change of a line, while the y-intercept is the point where the line crosses the y-axis. In the equation y = mx + b:
- m represents how steep the line is.
- b represents the starting value when x = 0.
9. Can you give a real-life example of slope?
A real-life example of slope is the steepness of a road or ramp. If a ramp rises 3 meters over a horizontal distance of 6 meters, the slope is 3/6 = 1/2. This means for every 1 unit forward, the height increases by 0.5 units. Slope is commonly used in construction, engineering, and physics to measure incline or rate of change.
10. What are common mistakes when finding slope?
Common mistakes when finding slope include subtracting coordinates incorrectly or dividing in the wrong order. To avoid errors:
- Keep the same order when subtracting: (y₂ − y₁) and (x₂ − x₁).
- Do not reverse only one pair of values.
- Watch for vertical lines where x₂ − x₁ = 0 (undefined slope).
- Simplify fractions correctly.
