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Slope Formula
Slope Formula is used to calculate a line’s inclination or steepness. It can be used to calculate the slope of any line by dividing the change in the yaxis by the change in the xaxis. The slope of a line is defined as the change in the line’s “y” coordinate with respect to the change in the line’s “x” coordinate. Finding the ratio of “vertical change” to “horizontal change” between any two distinct points on a line yields the slope. When the ratio is expressed as a quotient (“rise over run”), the same number is provided for every two distinct points on the same line. Learn more about slope formula in this article.
Quick Links
ToggleWhat is Slope?
Slope of a line determines its “steepness.” It is commonly represented by the letter m. As a result, the slope of a line is defined as the change in Y divided by the change in X. Because the change in Y is so large, the slope can be anything from zero to any number. However, the maximum slope is usually positive or negative infinity. Because the change in x is much smaller than the change in y, the change in x is much less than the change in y.
The slope of a line indicates how slanted the line is, comparing how much the line rises vertically to how much it runs horizontally.
Slope Formula
In coordinate geometry, the slope (often denoted as \( m \)) is calculated using the coordinates of two distinct points on the line. The formula for the slope of a line passing through points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ m = \frac{y_2 – y_1}{x_2 – x_1} \]
Where,
\( (x_1, y_1) \) and \( (x_2, y_2) \): These are the coordinates of two distinct points on the line.
\( y_2 – y_1 \): This represents the vertical change (rise) between the two points.
\( x_2 – x_1 \): This represents the horizontal change (run) between the two points.
Derivation of Slope Formula
The slope of a line is a measure of its steepness and is commonly denoted by the letter m. The slope can be derived using two points on the line. Here’s a stepbystep derivation of the slope formula:
Step 1. Identify two points on the line:
Let the coordinates of the two points be \((x_1, y_1)\) and \((x_2, y_2)\).
Step 2. Understand the change in coordinates:
The change in the xcoordinates (horizontal change) is given by \( \Delta x = x_2 – x_1 \).
The change in the ycoordinates (vertical change) is given by \( \Delta y = y_2 – y_1 \).
Step 3. Define the slope:
The slope \( m \) of the line is defined as the ratio of the vertical change to the horizontal change between the two points. Mathematically, this is written as:
\[
m = \frac{\Delta y}{\Delta x}
\]
Step 4. Substitute the changes in coordinates:
Substitute the expressions for \( \Delta y \) and \( \Delta x \) into the slope formula:
\[
m = \frac{y_2 – y_1}{x_2 – x_1}
\]
The slope formula is:
\[
m = \frac{y_2 – y_1}{x_2 – x_1}
\]
Slope Equation
The slope is calculated as the ratio of the yaxis change to the xaxis change. The slope of a straight line describes the angle of steepness from the horizontal, regardless of whether the line rises or falls. When the line does not rise or fall, its slope is zero. This is the case with a horizontal line, which extends indefinitely to the left or right but shows no indication of rise or fall.
The slope equation, often referred to as the equation of a line in slopeintercept form, is a way to express a linear equation using the slope of the line and its yintercept. The general form of this equation is:
y = mx + b
where:
y is the dependent variable (the value on the yaxis),
x is the independent variable (the value on the xaxis),
m is the slope of the line,
b is the yintercept (the point where the line crosses the yaxis).
To derive the slopeintercept form, we start from the basic definition of the slope and use a known point on the line.
Step 1: Start with the slope formula:
\[
m = \frac{y_2 – y_1}{x_2 – x_1}
\]
Step 2: Let \((x_1, y_1)\) be a specific point on the line, and \((x, y)\) be any other point on the line:
\[
m = \frac{y – y_1}{x – x_1}
\]
Step 3: Solve for \( y \):
\[
y – y_1 = m(x – x_1)
\]
This is the pointslope form of the equation of a line.
Step 4: Isolate \( y \) to convert to slopeintercept form:
\[
y = m(x – x_1) + y_1
\]
Simplifying further, we get:
\[
y = mx – mx_1 + y_1
\]
\[
y = mx + (y_1 – mx_1)
\]
Step 5: Identify the yintercept \( b \):
The term \( (y_1 – mx_1) \) represents the yintercept \( b \). Therefore, we can rewrite the equation as:
\[
y = mx + b
\]
How to Calculate Slope?
To determine the slope of a line, we only need two points from that line, (x1, y1) and (x2, y2). The slope of a straight line is calculated in three steps.
Step 1. Determine two points on the line.
Step 2: Choose one to be (x1,y1) and another to be (x2,y2).
Step 3: Calculate the slope using the Slope Formula.
Examples Using Slope Formula
Example 1: Find the slope of the line passing through the points (1, 2) and (4, 6)
Solution :
Identify the coordinates: \( (x_1, y_1) = (1, 2) \) and \( (x_2, y_2) = (4, 6) \).
Calculate the difference in yvalues: \( y_2 – y_1 = 6 – 2 = 4 \).
Calculate the difference in xvalues: \( x_2 – x_1 = 4 – 1 = 3 \).
Divide the differences: \( m = \frac{4}{3} \).
So, the slope of the line is \( \frac{4}{3} \).
Example 2: Find the slope of the line passing through the points \( (3, 5) \) and \( (2, 1) \).
Solution :
Identify the coordinates: \( (x_1, y_1) = (3, 5) \) and \( (x_2, y_2) = (2, 1) \).
Calculate the difference in yvalues: \( y_2 – y_1 = 1 – 5 = 6 \).
Calculate the difference in xvalues: \( x_2 – x_1 = 2 – (3) = 2 + 3 = 5 \).
Divide the differences: \( m = \frac{6}{5} \).
So, the slope of the line is \( \frac{6}{5} \).
Given two points (1, 2) and (4, 6).
Calculate the change in \( y \):
\[
\Delta y = 6 – 2 = 4
\]
Calculate the change in \( x \):
\[
\Delta x = 4 – 1 = 3
\]
Apply the slope formula:
\[
m = \frac{4}{3}
\]
So, the slope m of the line passing through the points (1, 2) and (4, 6) is \( \frac{4}{3} \)
FAQs (Frequently Asked Questions)
1. What is the significance of Slope Formula?
The Slope Formula is crucial for problemsolving. Every question based on the Slope Formula should be practised by students. It is also advised that when using the Slope Formula to solve questions, they concentrate on the chapter.
2. Can the slope formula be used for vertical lines?
No, the slope formula cannot be used for vertical lines because the difference in xcoordinates (
${}_{}$x2−x1) would be zero, leading to division by zero, which is undefined.
3. What is the slope of a horizontal line?
The slope of a horizontal line is zero because the difference in ycoordinates ($y_{2}−y_{1}$) is zero, making the numerator of the slope formula zero.
4. Can the slope formula be used in three dimensions?
The standard slope formula applies to two dimensions. In three dimensions, the concept of slope extends to direction ratios or gradients, involving vector calculus.