In mathematics, slope refers to a measure of the direction and steepness of a line. To determine the relationship between lines whether parallel, perpendicular, or at an angle – without assistance from the geometrical instrumentation, the most effective method is to calculate their respective slope.
The slope in mathematics of the line can be calculated by using two points on the line. In the upcoming text, we will delve into the concept of the slope with its respective mathematical formula and its types. In the example section, we will solve some examples to gain a better understanding.
Definition of Slope in Mathematics
The concept of slope in mathematics refers to the ratio of vertical change to the horizontal change between two points on a line or a plane, often denoted by the symbol “m”. The rate of change in vertical coordinates is represented as Δy, and the rate in horizontal coordinates is Δx. The slope of the line can be represented by tan θ.
m = tan θ = rate change in vertical coordinates/rate change in horizontal coordinates
m = tan θ = Δy / Δx
Slope also can be defined as the ratio of rise (change in the y-axis) to run (change in the x-axis).
m = slope = rise/run.
Formula of Slope in Mathematics
The formula for finding the slope between two points (x1, y1) and (x2, y2) that exist on the line is given below:
m = tan θ = vertical change / horizontal change
Slope = m = (y2 – y1) / (x2 – x1)
Types of Slope
There are four different types of slope in mathematics:
- Positive slope
- Negative slope
- Zero slope
- Undefined slope
1. Positive Slope
In a positive slope, the line goes up from left to right. In other words, by increasing the value of x, the value of y will be increased. The value of the positive slope lies between zeros and infinity.
2. Negative Slope
A negative slope in mathematics is one where the line moves from left to right. In other words, in negative slope value of y decreases by increasing x. The value of the negative slope lies between negative infinity and zero.
3. Zero Slope
If the line is parallel to the x-axis, the slope is 0 as there are no changes in y with increasing x. We will use the slope formula to understand it.
m = (y2 – y1) / (x2 – x1)
Since the rate of change in y is zero, so
m = 0 / (x2 – x1) =0
In zero slope, the angle along the x-axis will be zero or 180, so the slope,
m = arc tan (0) = 0
4. Undefined Slope
If the line is parallel to the y-axis, the value of y increases, but no change occurs in x. The slope will be undefined in this condition. I.e. m = (y2 – y1) / zero = undefined. In this condition, the angle with the x-axis will be 90, so as slope = m = arc tan (90) = undefined
Slope in Mathematics for Perpendicular Lines
The angle between perpendicular lines is always 90 degrees. The slope of perpendicular lines are negative reciprocals of each other, and the product of two or more perpendicular lines is negative. Mathematically if m1 and m2 are slopes of two line lines then m1×m2 = – 1
Determining the Slope of Parallel Lines
The angle inclination of parallel lines is the same. Let L1 and L2 be parallel lines along angles θ1 and θ2, respectively, along the x-axis. Such that θ1 = θ2, therefore the slope of given parallel lines is equal, i.e. m1 = m2
Solved Examples of Slope
Example 1
If you have two points on a line; (2, 4) and (6, 8), determine the slope and direction of this line.
Solution
Use the formula for slope i.e. m = (y2 – y1) / (x2 – x1)
Placing our given value in the above formula
Slope = m = 8 – 4 / 6 – 2
= 4 / 4 = 1
Hence slope = m = 1, which is positive, so the line goes up from left to right.
Example 2
Determine the slope of the line between two points (2, 3) and (-1, 2); also, find the inclination angle.
Solution
Use the formula for slope i.e. m = (y2 – y1) / (x2 – x1)
Substitute the given values in the slope formula
Slope = m = 2 – 3 / – 1 – 2
= – 1 / -3 = 1 / 3
The slope of the given points (2, 3) and (-1, 2) = 1 / 3
As tan θ = m, θ = tan-1 (m), as m is equal to 1 / 3 so put the value in the formula
θ = tan-1 (1 / 3)
θ = 0.2617 radian
Example 3
Calculate slope of given coordinate points: (1, 2) and (3, 4)
Solution
As,
m = (y2 – y1) / (x2 – x1)
Slope = m = 4 – 2 / 3 – 1
= 2 / 2 = 1
The slope of the given points (1, 2) and (3, 4) is 1.
Conclusion
In this article, we have described slope with its formula. We have covered different types of slopes, which are positive slope, negative slope, zero slope, and undefined slope. We learned how to determine the slope between two given points with formulas by examples. After understanding this article better, you can find a slope between two given points.