Slope = \( \frac{change \;in\; y-coordinates}{change \;in\; x-coordinates} \) = \( \frac{{{y} _ {2}} \;-\; {{y} _ {1}}}{{{x} _ {2}} \;-\; {{x} _ {1}}} \)
The slope of a line is the same, no matter what two points on the line are used.
y = mx + b is an equation in slope-intercept form. The slope is m and b is the y-intercept.
In mathematics, slope is expressly referred to as the steepness of a line. Inclination is a synonym for steepness. Slope is a number that represents a ratio describing the incline of a line. It is calculated by finding the ratio of the vertical change (rise) to the horizontal change (run).
Slope = Rise/Run
Rise is how much a line goes up or down.
Run is how much a line moves left or right.
Slope ⅔ has a rise of 2 and a run of 3.
Since the rise is represented by a positive number, it means that the line is “rising vertically” or “going up.”
Since the run is represented by a positive number, it means that the line is “running horizontally” or “going to the right.”
Let us start at the point (-6, 0) and plot a line with a slope of ⅔. We get the following graph below:
A slope of ⅔ means that every time the line rises 2, it runs 3.
Since the slope is a ratio, it can be expressed as various ratios of numbers.
Types of slopes:
Positive Slope: A line that has a positive slope rises from left to right. A line with a positive slope increases upward from left to right. The equation y = ½x + 2 has a positive slope of ½.
Negative Slope: A line that has a negative slope falls from left to right. A line with a negative slope decreases downward from left to right. The equation y = -4x + 2 has a negative slope of -4.
Horizontal Line Slope: A line with zero slope is horizontal because the rise is 0, and 0 divided by any non-zero number is 0. A horizontal line has a slope of zero because it does rise vertically. A horizontal line is a straight line that is mapped from left to right and it is parallel to the x-axis in the plane coordinate system. It does not rise or fall as it moves along the x-axis. The equation y = 3 has no slope, so the slope is 0. See below:
Vertical Line Slope: A line with an undefined slope is vertical because the run is 0, and any non-zero number divided by 0 is undefined. Vertical lines have no slope, as they do not have any steepness. We cannot define the steepness of a vertical line.
Parallel Line Slope: The slope of parallel lines is always equal. Parallel lines have the same slope because their rise over run ratio is equal. They make an equal angle with the positive x-axis. The following graph is a parallel line slope with equations y = 2x + 2 and y = 2x - 3:
Perpendicular Line Slope: The slopes of two perpendicular lines are negative reciprocals of each other. Simply put, the product of the slopes of two perpendicular lines equals -1.The following graph shows a perpendicular line slope with equations y = -10x + 1 and y = -1/-10x + 1:
