What is Slope of a Line?
A slope indicates steepness as well as direction. The slope of a line can be positive or negative. When a line has a positive slope, it moves upward from left to right. When moving from left to right, a line with a negative slope moves down. Two linear functions are parallel if their slopes are the same.
The ratio of the “vertical change” to the “horizontal change” between (any) two distinct points on a line is used to calculate the slope.
The ratio is sometimes expressed as one (“rise over run”) which gives the same number for any pair of two distinct points on the same line.
Skiers and snowboarders must consider the slopes of hills when judging danger, speed, and other factors.
How to find the Slope of a Line using a Graph?
The slope ‘m’ of a line is equal to the ratio of the change in y-coordinates to the change in x-coordinates between any two points on the line. Here, the change in y-coordinates is called the rise, and the change in x-coordinates is called the run. So, the slope is the ratio of rise to run.
The slope of a line is a measurement of its steepness.
A slope can be determined as follows:
Step 1. Determine the coordinates of any two points on the line.
Step 2. Calculate the difference between the corresponding values of the y-coordinate (called rise).
Step 3. Calculate the difference between the corresponding values of the x-coordinate (called run).
Step 4. Divide the y-coordinate difference by the x-coordinate difference (or ratio of rise to run).
Slope, \(\text{m} =\frac{\text{Rise}}{\text{Run}}\)
\(\text{m} =\frac{\text{Change in y}}{\text{Change in y}}\)
\(\text{m} =\frac{y_2-y_1}{x_2-x_1}\)
where \((x_1,y_1)\) and \((x_2,y_2)\) are the coordinates of the two points lying on the line.
Positively sloped lines rise from left to right and negatively sloped lines fall from left to right.
The Slope of Horizontal and Vertical Lines
A horizontal line has a zero slope, which means it does not rise or fall as we move from left to right.
Also, the slope of vertical lines appears to be an infinitely large, undefined value, so they are said to have an “undefined slope.”
Example: Find the slope of a line at y = 4.
Solution:
It is given that y = 4. So, its graph will be a horizontal line that will cut the y-axis on the coordinate 4.
The following graph depicts this solution:
Hence, the slope is o.
Example: Find the slope of a line at x = 4.
Solution:
It is given that x = 4. So, its graph will be a vertical line that will cut the x-axis at the coordinate 4.
The following graph depicts this solution:
Hence, the slope will be undefined.
The Slope of Line through given Points
To find the slope of a line when the coordinates of two points on the line are given, use the formula for slope.
\(\text{m}=\frac{y_2-y_2}{x_2-x_1}\), that is, the change in the values of y over the change in the values of x.
Here, \(x_1\) and \(y_1\) are the coordinates of the first point, and the coordinates of the second point are \(x_2\) and \(y_2\).
Example: Find the slope of the line passing through the two given points A(2,6) and B(-3,-4).
Solution:
We have the coordinates A(2, 6) and B(- 3, – 4).
The slope of the line passing through two points is given as \(\frac{y_2-y_2}{x_2-x_1}\), that is,
\(=\frac{-4-6}{-3-2}\) Take \((x_1,y_1)\) as (2, 6) and \((x_2,y_2)\) as (- 3, – 4)
\(=\frac{-10}{-5}\)
\(=\frac{10}{5}\)
\(= 2\)
Parallel Lines
Parallel lines are those that are equidistant from each other and never meet, no matter how far they are extended in either direction.
The slope of parallel lines
The slope of parallel lines is the same or equal.
\(m_1=m_2\)
Here, \(m_1\) and \(m_2\) represent the slope of two different and parallel lines.
Example: Find the slope of a line parallel to the line 2x + y = 3.
Solution:
2x + y = 3 Equation of the parallel line
y = – 2x + 3 Equation in slope – intercept form
The slope of any parallel line to the line y = – 2x + 3 is – 2.
Solved Slope of the Lines Examples
Example 1: Describe the slope of the line and find its slope.
Solution: The line rises from left to right. The slope is thus positive. Find the rise and the run of the line using the graph.
The rise and run are calculated by subtracting the coordinates of the x-axis (for the run) and the y-axis (for the rise).
Slope, \(\text{m}=\frac{y_2-y_1}{x_2-x_1}\)
Take \((x_1,~y_1)\) as (-2,-2) and \((x_2,~y_2)\) as (3,2)
\(\text{m}=\frac{\text{Rise}}{\text{Run}}\)
\(\text{m}=\frac{2-(-2)}{3-(-2)}\)
\(\text{m}=\frac{4}{5}\)
Example 2: Describe the slope of the line and find its slope.
Solution: The line falls from left to right. The slope is thus negative. Find the slope of the line using the coordinates of the two points lying on the line.
Slope, \(\text{m}=\frac{y_2-y_1}{x_2-x_1}\)
Take \((x_1,y_1)\) as (3, 1) and \((x_2,y_2)\) as (- 4, – 3)
\(\text{m}=\frac{-3-1}{-4-3}\)
\(\text{m}=\frac{-4}{-7}\)
\(\text{m}=\frac{4}{7}\)
Hence, the slope is \(\text{m}=\frac{4}{7}\).
Example 3: Find the slope of the line passing through the two given points C(1, 4) and D(- 5, – 2).
Solution: We have the coordinates C(1, 4) and D(- 5, – 2).
The slope of the line passing through two points is given as \(\frac{y_2-y_1}{x_2-x_1}\), that is,
\(=\frac{- 2 – 4}{- 5 – 1}\) [Take \((x_1,y_1)\) as (1, 4) and \((x_2,y_2)\) as (- 5, – 2)]
\(=\frac{-6}{-6}\)
\(=\frac{6}{6}\)
\( = 1 \)
Hence, the slope is 1.
Example 4: Find the slope of a line parallel to the line 7x + y = 6.
Solution:
2x + y = 3 Equation of the parallel line
y = – 7x + 6 Equation in the slope-intercept form
The slope of this line is – 7.
Hence, the slope of any line parallel to the line y = – 7x + 6 is – 7.
Example 5: The table depicts the distance y (in miles) travelled by a car from a toll plaza x minutes after staff change. The graph of the table’s points forms a straight line on the coordinate plane. Find and interpret the slope of the line.
x | 2 | 4 | 5 | 7 |
|---|---|---|---|---|
y | 12 | 8 | 4 | 2 |
Solution:
Use the formula of slope by taking any two points from the table:
\(\text{m}=\frac{y_2-y_1}{x_2-x_1}\) Formula for the slope of a line
\(\text{m}=\frac{4-8}{5-4}\) [Take \((x_1,y_1)\) as (4, 8) and \((x_2,y_2)\) as (5,4)]
\(\text{m}=\frac{-4}{1}\) or – \( 4 \) Simplify
The slope is – 4. So, the distance between the car and the toll plaza decreases by 4 miles per minute.
A positive slope is a line where changes in the x and y coordinates always have the same sign and move upwards from left to right.
A negative slope is a line where the changes in the x and y coordinates always have a different sign and move downwards from left to right.
The undefined slope is the slope of a vertical line. For such a line, the x-coordinates remain constant regardless of the change in the y-coordinates. Vertical lines do not run left or right. Instead, they rise or fall straight up or down. The slope is the ratio of changes in the y-coordinate to changes in the x-coordinate, which is undefined for a vertical line.
Parallel lines have the same slope. Two lines are parallel if their slopes are equal and their y-intercepts are different.