What is the Slope Formula?
Slope of a line can be defined as the value of ratio of change in y (the rise) to the change in x (the run) between any two points of a line \((x_1,~y_1)\) and \((x_2,~y_2)\). The slope of a line can also be defined as the measurement of the steepness of that line. The slope of a line is generally denoted by ‘m’.
Slope, m = \(
= [latex\frac{Change~in~y}{Change~in~x}\)
= \(\)
Types of Slope
Positive Slope: A line with a positive slope rises from left to right.
Negative Slope: A line with a negative slope rises from right to left.
Zero slope: A horizontal line has zero slope.
Undefined slope: The slope of a vertical line is ‘undefined’ or ‘not defined’.
Solved Examples
Example 1: Find the slope of a line passing through the points (3, 7) and (5, 9).
Solution:
Slope, m = \(
= \(\)y_2\), -1 for \(y_1\), 4 for \(x_2\) and 7 for \(x_1\)
= \( Simplify
= -[latex\frac{9}{3}\)
= – 3
Hence, the slope of the given line is – 3 (Negative slope).
Example 3:
After observing the graph of a line, Rhea realized that the rise of the graph is 12 units and the run is 4 units. Find the slope of Rhea’s line?
Solution:
Given that, Rise = 12 units
Run = 4 units.
Slope, m = \(
m = [latex\frac{12}{4}\) = 3.
Hence, the slope of the line is 3
Improper fractions are fractions that have numerators that are greater than or equal to the denominators. For example: \(\frac{23}{5},\frac{13}{11},\frac{7}{2},\frac{6}{6}\) and so on.
Mixed numbers are numbers between two whole numbers. So mixed numbers have a whole number part and a proper fraction. For example, \(1\frac{2}{3}\) is a mixed number. This number lies between the wholes, 1 and 2. In \(1\frac{2}{3}\), 1 is the whole number and \(\frac{2}{3}\) is the proper fraction.
First, we need to convert the mixed numbers into improper fractions. Then the normal multiplication process followed for fractions can be applied. The numerators are multiplied, then the denominators, and the respective products are expressed as a fraction, which is the answer.
The mixed number is first converted to an improper fraction. Then the numerator and the whole number is multiplied, the product becomes the numerator of the required fraction and the denominator is the same as that of the improper fraction. Thus, the required fraction is obtained.
A proper fraction has the numerator lesser than the denominator. For example, \(\frac{1}{2},\frac{3}{4},\frac{11}{13}\) and so on.