Slope Formula

The slope of a line calculates the "steepness" of a line. It is usually denoted by the letter m. So, the slope of a line is the change in Y divided by the change in X. As the change in Y is very high, the slope can range from zero to any number that we can think of. However, we usually have a maximum slope of positive or negative infinity. The change in x is much smaller than the change in y, which means that the change in x is much less than the change in y.

The slope of a line shows how slant the line is, how much the line rises vertically is compared with how much it runs horizontally. Being able to find the slope of a line, or using the slope to find points on the line, is an important skill used in economics, geoscience, accounting/finance, and other fields. The slope of a line is also defined as the ratio of rise over run. 

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The x and y coordinates of the lines are used to calculate the slope of the lines. It is the ratio of the change in the y-axis to the change in the x-axis.

The formula to calculate slope is given as,

That is, 

\[\frac{\text{change in y Coordinates}}{\text{change in x Coordinates}}\]

Where m is the slope of the line. x1, x2 are the coordinates of x-axis and y1, y2 are the coordinates of y-axis.

The slope of the line can be a positive or negative value.

x and y are only used to identify the two points. They are not values or exponents, the points can be given any names.

\[m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\]

Slope formula when the general equation of a straight line is given

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If the general equation of a straight line is given as

ax + by + c  =  0,   

then, the formula for the slope of the line is 

m = - coefficient of x / coefficient of y 

m = -a/b

Slope Intercept Formula

Linear equations are “straight line” equations that have simple variable expressions with terms without exponents on them. You are dealing with a straight line equation, if you come across an equation with only x and y. To find the equation of a line and y-intercept in the steepness of the line, we use slope intercept formula.

Slope intercept formula is

The values used in formula are as follows:

• m is the slope of the line

• b is the y-intercept of the line

Rules for Calculating the Slope of Line

Method 1:

Step 1: Find two points on the line.

Step 2: Count the rise. The number of units counts up or down to get from one point to the next. Record this number as your numerator.

Step 3: Count the run. The number of units counts left or right to get to the point. Record this number as your denominator.

Step 4: Simplify your fraction if possible.

Method 2:

To find out the slope of a line, we need only two points from that line, (x1, y1) and (x2, y2). 

There are three steps for calculating the slope of a straight line.

Step1: Identify two points on the line.

Step2: Select one to be (x1,y1) and the other to be(x2,y2).

Step3: Use the slope of the line formula to calculate the slope.

Some of the important points to remember to find the slope of the line. They are as follows:

• The slope formula can give a positive or negative result. 

• If the slope is a positive value, the line is in a rising state. 

• If the slope is a negative value, the line is descending. 

• Vertical lines have no slope.

• Horizontal lines have a zero slope.

• Parallel lines have equal slopes. 

• Perpendicular lines have negative reciprocal slopes. 

Solved Examples

Example 1: Find the slope of the line whose coordinates are (2,6) and (5,1).

Solution:

We have,

(x1, y1) = (2, 6)

(x2, y2) = (5, 1)

The slope of a line formula is \[m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\]

m = (1 − 6/ 5 − 2)

m = −5/3

m = − 1.666

Example 2: If the slope of a line passing through the points (4, x) and (2, -7) is 3, then what is the value of x?

Solution:

We have

Slope = m = 3

Points:

(x1, y1 ) = (4, x)

(x2, y2) = (2, -7)

We know that,

Slope \[m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\]

3 = (-7 - x)/(2 – 4)

3 = (-7 - x)/(-2)

or, -7 – x = 3(-2)

or, -7 – x = -6

so, x = -7 + 6 = -1

Therefore, the value of x = -1.