Slope
- What is slope?
- The Slope of a Line
- Types of Slope
- Slope Between Two Lines
- The Slope of a Horizontal Line
- The Slope of a Vertical Line
- The Slope of a Perpendicular Line
- The Slope of a Parallel Line
- Finding Slope from a Graph
- Practice Problems
- Frequently Asked Questions
What is slope?
The slope is a fundamental concept in mathematics that measures the steepness of a line. It represents how much a line rises or falls as you move along it. To find the slope of a line, you need to compare the vertical change (rise) to the horizontal change (run) between two points on the line. In Maths, the slope is usually represented by the letter `‘m’`.
The slope of a line is the change in `y`-coordinate with respect to the change in `x`-coordinate.In simpler terms, slope tells us how much a line slants upwards or downwards.
The Slope of a Line
The slope of a line is a measure of its steepness or incline. It represents the rate at which the line rises or falls as you move along it. This can be expressed as the ratio of the change in `y`-coordinates (rise) to the change in `x`-coordinates (run).
This can be calculated using the formula:
\[ \text{Slope} = \frac{{\text{Change in } y}}{{\text{Change in } x}} \]
This formula compares how much the line rises or falls (change in `y`) relative to how much it moves horizontally (change in `x`) between two points on the line.
For example, if you have two points `(x_1, y_1)` and `(x_2, y_2)` on a line, the slope can be found by dividing the difference in the `y`-coordinates by the difference in the `x`-coordinates:
\[\text{Slope}= \frac{y_2 - y_1}{x_2 - x_1}\]
Example: Find the slope of the line passing through the points \( (2, 3) \) and \( (5, 9) \).
Solution:
To find the slope, we first need to calculate the change in `y` and the change in `x` between the two points.
Change in `y =` \( 9 - 3 = 6 \)
Change in `x =` \( 5 - 2 = 3 \)
Now, we can use the slope formula:
\( \text{slope} = \frac{{6}}{{3}} = 2 \)
So, the slope of the line passing through the points \( (2, 3) \) and \( (5, 9) \) is `2`.
Types of Slope
The slope of a straight line can be either positive, negative, zero or undefined.
Positive Slope:
When a line rises from left to right, it has a positive slope. In other words, as you move along the line from left to right, the `y`-values increase as `x` increases. An example of a positive slope is a line with a slope of `2`, where for every `1` unit increase in `x`, the `y`-value increases by `2` units.
Negative Slope:
When a line falls from left to right, it has a negative slope. This means that as you move along the line from left to right, the `y`-values decrease as `x` increases. For instance, a line with a slope of `-3` indicates that for every `1` unit increase in `x`, the `y`-value decreases by `3` units.
Zero Slope:
When a line is horizontal, meaning it neither rises nor falls, it has a zero slope. In this case, the `y`-values remain constant regardless of changes in `x`. For example, a line with a slope of `0` means that no matter how much `x` changes, `y` remains the same.
When a line is vertical, it has an undefined slope. Vertical lines have no slope, as they do not have any steepness. In this case, the `x`-values remain constant regardless of changes in `y`. We cannot define the steepness of vertical lines.
Slope Between Two Lines
The slope between two lines is a measure of their relative steepness or inclination with respect to each other. It tells you how much one line is slanted compared to the other. To find the slope between two lines, you can select any two points, one from each line, and then calculate the slope using the formula:
\[\text{Slope} = \frac{\text{Change in }y}{\text{Change in }x}\]
This helps determine the angle or degree of tilt between the two lines.
The Slope of a Horizontal Line
A horizontal line runs parallel to the `x`-axis and has a slope of zero. This means that no matter how far you move along the line horizontally (in the `x`-direction), the vertical `(y)` coordinate remains constant. In other words, the line neither rises nor falls as you move along it.
Example: Find the slope of the line \( y = 3 \).
Solution:
Since the line is horizontal, it does not rise or fall; therefore, its slope is zero.
The Slope of a Vertical Line
A vertical line runs parallel to the `y`-axis. Unlike horizontal lines, vertical lines have undefined slopes because they have no horizontal change (run). This means that as you move along the line vertically (in the `y`-direction), the horizontal `(x)` coordinate remains constant. In other words, the line does not move left or right.
Example: Determine the slope of the line \( x = -2 \).
Solution:
Since the line is vertical, it has an undefined slope.
The Slope of Perpendicular Lines
Perpendicular lines are two lines that intersect at a `90°` angle and have slopes that are negative reciprocals of each other. In other words, if the slope of one line is \( m \), then the slope of a line perpendicular to it is `-1/m`.
Example: Given the line \( y = 2x + 3 \), find the equation of a line perpendicular to it passing through point \( (1, -1) \).
Solution:
First, find the slope of the given line. Since it's in the form \( y = mx + b \), the slope is \( m = 2 \).
The slope of the perpendicular line would be \( -\frac{1}{2} \).
Next, use the point-slope form \( y - y_1 = m(x - x_1) \) and substitute \( m = -\frac{1}{2} \) and \( (x_1, y_1) = (1, -1) \).
This gives us the equation of the perpendicular line as
\( y - (-1) =-\frac{1}{2}(x - 1) \)
\( y + 1 = -\frac{1}{2}(x - 1) \)
The Slope of Parallel Lines
Parallel lines are two lines that never intersect and have the same slope. This means that they have the same steepness and inclination, even though they may be located at different positions on the coordinate plane.
Example: If the equation of one line is \( y = 4x - 2 \), find the equation of a parallel line passing through the point \( (2, 5) \).
Solution:
The given line has a slope of \( m = 4 \). Since parallel lines have same slope, the slope of the parallel line is also \( m = 4 \).
Next, use the point-slope form \( y - y_1 = m(x - x_1) \) and substitute \( m = 4 \) and \( (x_1, y_1) = (2, 5) \).
This gives us the equation of the parallel line as
\( y - 5 = 4(x - 2) \)
Finding Slope from a Graph
To find the slope of a line from a graph, one should follow these steps:
Step `1`: Select two points: Identify two points on the line. These points should be clear and easily identifiable on the graph. We call them lattice points.
Step `2`: Determine coordinates: Note down the coordinates of the two points you've selected. You'll need the \( (x, y) \) coordinates of each point.
Step `3`: Calculate the change in `y` and change in `x`: Find the difference between the \( y \)-coordinates of the two points (change in \( y \)) and the difference between the \( x \)-coordinates of the two points (change in \( x \)).
Step `4`: Apply Slope Formula: \( \text{slope} = \frac{{\text{change in } y}}{{\text{change in } x}} \).
Step `5`: Interpret the Result: The value you calculate is the slope of the line.
Example:
In the given graph, let's select two points: \( A \) with coordinates \( (2, 4) \) and \( B \) with coordinates \( (6, 10) \).
Now, let's calculate the change in \( y \) and the change in \( x \):
Change in \( y = 10 - 4 = 6 \)
Change in \( x = 6 - 2 = 4 \)
Next, we'll apply the slope formula:
\( \text{slope} = \frac{{\text{change in } y}}{{\text{change in } x}} = \frac{6}{4} = 1.5 \)
So, the slope of the line passing through points \( A \) and \( B \) is \( 1.5 \).
Practice Problems
Q`1`. Given two points \( (3, 5) \) and \( (7, 11) \), what is the slope of the line passing through these points?
- `2`
- `1.5`
- `1`
- `0.5`
Answer: b
Q`2`. If a line has a slope of \( -3 \), which of the following statements is true?
- The line rises as it moves from left to right.
- The line falls as it moves from left to right.
- The line is horizontal.
- The line is vertical.
Answer: b
Q`3`. Which of the following pairs of lines are perpendicular?
- \( y = 2x + 3 \) and \( y = -2x - 3 \)
- \( y = 3x + 2 \) and \( y = -3x + 2 \)
- \( y = \frac{1}{2}x + 4 \) and \( y = -2x + 4 \)
- \( y = 5x + 1 \) and \( y = \frac{1}{5}x + 1 \)
Answer: c
Q`4`. If the slope of a line is \( \frac{3}{4} \), what is the slope of a line parallel to it?
- \( -\frac{3}{4} \)
- \( \frac{4}{3} \)
- \( \frac{3}{4} \)
- \( -\frac{4}{3} \)
Answer: c
Q`5`. Which of the following lines has an undefined slope?
- \( y = 2x + 3 \)
- \( y = -4 \)
- \( y = 5 \)
- \( x = 3 \)
Answer: d
Frequently Asked Questions
Q`1`. What is the slope in mathematics?
Answer: Slope is a measure of how steep a line is. It indicates the rate at which the line rises or falls as you move along it.
Q`2`. How do you calculate slope?
Answer: Slope is calculated by dividing the change in `y`-coordinates by the change in `x`-coordinates between two points on a line. This is represented by the formula:
\[ \text{Slope} = \frac{{\text{Change in } y}}{{\text{Change in } x}} \]
Q`3`. What does a positive slope represent?
Answer: A positive slope indicates that the line rises from left to right. As you move along the line from left to right, the `y`-values increase.
Q`4`. What does a negative slope indicate?
Answer: A negative slope indicates that the line falls from left to right. As you move along the line from left to right, the `y`-values decrease.
Q`5`. What does a slope of zero mean?
Answer: A slope of zero means that the line is horizontal. It neither rises nor falls as you move along it.
Q`6`. What is the significance of perpendicular lines in terms of slope?
Answer: Perpendicular lines have slopes that are negative reciprocals of each other. In other words, if one line has a slope of \( m \), a line perpendicular to it will have a slope of ` -1/m`.
Q`7`. How are parallel lines related in terms of slope?
Answer: Parallel lines have the same slope. They may be located at different positions on the coordinate plane but maintain the same steepness throughout.