THE SLOPE FORMULA

There is a formula you can use to find the slope of a line, which is usually represented by the letter m. To use this formula, you need the coordinates of two different points on the line.

Words : 

The slope of a line is the ratio of the difference in y-values to the difference in x-values between any two different points on the line.

Formula : 

If (x₁, y₁) and (x₂, y₂) are any two different points on a line, the slope of the line is 

Formula : 

If (2, -3) and (1, 4) are two points on a line, the slope of the line is

m  =  [4 - (-3)]/(1 - 2)

=  (4 + 3)/(-1)

=  7/(-1)

=  -7

Finding Slope by Using the Slope Formula

Example 1 : 

Find the slope of the line that contains (5, -3) and (-2, 3).

Solution :

Use the slope formula.

m  =  (y₂ - y₁)/(x₂ - x₁)

Substitute (4, -2) for (x₁ , y₁) and (-1, 2) for (x₂,  y₂).

m  =  [3 - (-3)]/(-2 - 5)

m  =  (3 + 3)/(-7)

m  =  6/(-7)

m  =  -6/7

The slope of the line that contains (5, -3) and (-2, 3) is -6/7.

Finding Slope from Graphs and Tables

Example 2 : 

The graph shows a linear relationship. Find the slope.

Solution :

Let (-2, -1) be (x₁, y₁) and (2, 2) be (x₂ , y₂).

Use the slope formula.

m  =  (y₂ - y₁)/(x₂ - x₁)

Substitute (-2, -1) for (x₁ , y₁) and (2, 2) for (x₂,  y₂).

m  =  [2 - (-1)]/[2 - (-2)]

m  =  (2 + 1)/(2 + 2)

m  =  3/4

The slope of the line is 3/4.

Example 3 : 

The table shows a linear relationship. Find the slope.

Solution :

Choose any two points from the table.

Let (2, 0) be (x₁, y₁) and (2, 3) be (x₂ , y₂).

Use the slope formula.

m  =  (y₂ - y₁)/(x₂ - x₁)

Substitute (2, 0) for (x₁ , y₁) and (2, 3) for (x₂,  y₂).

m  =  (3 - 0)/(2 - 2)

m  =  3/0

m  =  Undefined

The slope is undefined.

Environmental Science Application

Example 4 : 

The graph shows how much water is in a reservoir at different times. Find the slope of the line. Explain what the slope represents.

Solution :

Use the slope formula.

m  =  (y₂ - y₁)/(x₂ - x₁)

Here, 

(x₁ , y₁)  =  (20, 3000)

(x₂,  y₂)  =  (60, 2000)

Then, 

m  =  (2000 - 3000)/(60 - 20)

m  =  -1000/40

m  =  -25

The slope is -25.

In the given situation, y represents volume of water and x represents time.

So slope represents

change in volume/change in time

in units of

thousands of cubic feet/hours.

A slope of -25 means the amount of water in the reservoir is decreasing (negative change) at a rate of 25 thousand cubic feet each hour.

Finding Slope from an Equation

Example 5 :

Find the slope of the line described by 6x - 5y = 30.

Solution : 

The line contains (5, 0) and (0, -6). Use the slope formula.

Use the slope formula.

m  =  (y₂ - y₁)/(x₂ - x₁)

Substitute (5, 0) for (x₁ , y₁) and (0, -6) for (x₂,  y₂).

m  =  (-6 - 0)/(0 - 5)

m  =  -6/(-5)

m  =  6/5

The slope is 6/5.

Example 6 :

Felicia Johnson paid $125 to join a tennis club. She pays an additional $5 every time she uses one of the club's tennis courts. Use this information to answer the following questions.

a. Write an equation that describes Felicia's total cost for playing tennis as a function of the number of times she plays. Let C = the total cost and n = the number of times she plays.

b. Describe the domain and range of the function.

c. Felicia does not want to spend more than $275 to play tennis during the summer.

What is the maximum number of times that she can play tennis on the club's courts for this amount?

Solution :

a) C = 125 + 5n

b) Domain is all real numbers and range is 125, 130, 135, .........

c)  When C =  275

275 = 125 + 5n

5n = 275 - 125

5n = 150

n = 150/5

n = 30

So, he can play 30 times.

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