Slope of a line
Slope of a Line
The slope of a line is a measure of its steepness and direction. In mathematics, particularly in coordinate geometry, the slope is a crucial concept when dealing with straight lines. It is often denoted by the letter 'm'.
Definition
The slope of a line is defined as the ratio of the change in the y-coordinate (rise) to the change in the x-coordinate (run) as we move from one point to another along the line. Mathematically, if we have two points on a line, $(x_1, y_1)$ and $(x_2, y_2)$, the slope m is given by:
$$ m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} $$
Types of Slopes
Depending on the direction and steepness, slopes can be categorized as follows:
| Slope Type | Description | Example |
|---|---|---|
| Positive | Line rises from left to right | $y = 2x + 1$ |
| Negative | Line falls from left to right | $y = -2x + 1$ |
| Zero | Horizontal line | $y = 3$ |
| Undefined | Vertical line | $x = 4$ |
Calculating the Slope
Example 1: Positive Slope
Consider two points on a line, A(1, 2) and B(3, 6). To find the slope of the line passing through these points:
$$ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2 $$
The slope is 2, which means the line rises 2 units for every 1 unit it moves to the right.
Example 2: Negative Slope
Consider two points on a line, C(-1, 4) and D(2, -1). To find the slope:
$$ m = \frac{-1 - 4}{2 - (-1)} = \frac{-5}{3} = -\frac{5}{3} $$
The slope is $-\frac{5}{3}$, indicating the line falls 5 units for every 3 units it moves to the right.
Example 3: Zero Slope
For a horizontal line such as $y = 5$, regardless of the x-values, the y-value remains constant. Therefore, the slope is:
$$ m = \frac{\Delta y}{\Delta x} = \frac{0}{\Delta x} = 0 $$
Example 4: Undefined Slope
For a vertical line such as $x = -2$, the x-value remains constant, but the y-value can be any number. This results in a division by zero when calculating the slope:
$$ m = \frac{\Delta y}{0} $$
Since division by zero is undefined, the slope of a vertical line is also undefined.
Slope-Intercept Form
The slope-intercept form of a line's equation is $y = mx + b$, where m is the slope and b is the y-intercept (the y-coordinate of the point where the line crosses the y-axis).
Example 5: Slope-Intercept Form
Given the equation $y = -3x + 7$, the slope m is -3, indicating a negative slope, and the y-intercept b is 7.
Point-Slope Form
Another way to express the equation of a line is the point-slope form, which is useful when we know a point on the line $(x_1, y_1)$ and the slope m:
$$ y - y_1 = m(x - x_1) $$
Example 6: Point-Slope Form
Given a point (2, 3) and a slope of 4, the equation of the line is:
$$ y - 3 = 4(x - 2) $$
Parallel and Perpendicular Lines
- Parallel lines have the same slope.
- Perpendicular lines have slopes that are negative reciprocals of each other.
Example 7: Parallel Lines
If line L1 has a slope of 5, then any line L2 that is parallel to L1 also has a slope of 5.
Example 8: Perpendicular Lines
If line M1 has a slope of $\frac{3}{4}$, then a line M2 that is perpendicular to M1 will have a slope of $-\frac{4}{3}$.
Conclusion
Understanding the slope of a line is fundamental in coordinate geometry. It allows us to determine the direction and steepness of a line, write equations of lines, and analyze the relationships between multiple lines, such as whether they are parallel or perpendicular. Remember that the slope is a ratio that can be positive, negative, zero, or undefined, and each type of slope has its own geometric interpretation on the Cartesian plane.