FIND THE SLOPE FROM AN EQUATION

We can find the slope of a line from its equation using either of two methods explained below.  

Method 1 : 

Slope intercept form equation of a line :

y  =  mx + b

We can write the equation of a line in slope intercept form and find the slope 'm' which is being as the coefficient of x.

Method 2 : 

General form equation of a line :

y  =  mx + b

If the equation of a line is given general form, we can find the slope of the line using the formula given below.

slope (m)  =  - coefficient of x / coefficient of y

Example 1 :

Find the slope of the line.  

x - y + 1  =  0

Solution : 

Method 1 :

Write the equation in slope intercept form.

x - y + 1  =  0

Add y to each side.

x + 1  =  y

(or)  y  =  x + 1

The above line is in slope intercept form. 

Comparing y = mx + b and y = x + 1, 

slope (m)  =  1

Method 2 :

Given equation is already in general form.

x - y + 1  =  0

slope (m)  =  - coefficient of x / coefficient of y

slope (m)  =  - 1 / (-1)

slope (m)  =  1

Example 2 :

Find the slope of the line. 

5x - 3y  =  0

Solution :

Method 1 :

Write the equation in slope intercept form.

5x - 3y  =  0

Add 3y to each side.

5x  =  3y

Divide each side by 3.

5x/3  =  y

(or)  y  =  5x / 3

The above line is in slope intercept form. 

Comparing y = mx + b and y = 5x/3, 

slope (m)  =  5/3

Method 2 :

Given equation is already in general form.

5x - 3y  =  0

slope (m)  =  - coefficient of x / coefficient of y

slope (m)  =  - 5 / (-3)

slope (m)  =  5/3

Example 3 :

Find the slope of the line.

4x + 1  =  2y

Solution :

Method 1 :

Write the equation in slope intercept form.

4x + 1  =  2y

Divide each side by 2.

(4x + 1) / 2  =  y

4x/2 + 1/2  =  y

2x + 1/2  =  y

(or)  y  =  2x + 1/2

The above line is in slope intercept form. 

Comparing y = mx + b and y = 2x + 1/2, 

slope (m)  =  2

Method 2 :

Write the equation in general form. 

4x + 1  =  2y

Subtract 2y from each side. 

4x - 2y + 1  =  0

slope (m)  =  - coefficient of x / coefficient of y

slope (m)  =  - 4 / (-2)

slope (m)  =  2

Example 4 :

Find the slope of the line.

3y - 2x + 9  =  0

Solution :

Method 1 :

Write the equation in slope intercept form.

3y - 2x + 9  =  0

Add 2x to each side.

3y + 9  =  2x

Subtract 9 from each side. 

3y  =  2x - 9

Divide each side by 3.

y  =  (2x - 9) / 3

y  =  2x/3 - 9/3

y  =  2x/3 - 3

The above line is in slope intercept form. 

Comparing y = mx + b and y = 2x/3 - 3, 

slope (m)  =  2/3

Method 2 :

Write the equation in general form.

3y - 2x + 9  =  0

-2x + 3y + 9  =  0

slope (m)  =  - coefficient of x / coefficient of y

slope (m)  =  - (-2) / 3

slope (m)  =  2/3

Example 5 :

Find the slope of the line.

y - 4  =  -3(x - 3)

Solution :

Method 1 :

Write the equation in slope intercept form.

y - 4  =  -3(x - 3)

Distributive property. 

y - 4  =  -3x + 9

Add 4 to each side.

y  =  -3x + 13

The above line is in slope intercept form. 

Comparing y = mx + b and y = -3x + 13, 

slope (m)  =  -3

Method 2 :

Write the equation in general form.

y - 4  =  -3(x - 3)

Distributive property. 

y - 4  =  -3x + 9

Add 3x to each side. 

3x + y - 4  =  9

Subtract 9 from each side. 

3x + y - 13  =  0

slope (m)  =  - coefficient of x / coefficient of y

slope (m)  =  - 3 / 1

slope (m)  =  -3

Example 6 :

Find the slope of the line.

y - 8  =  (-1/2)(x + 4)

Solution :

Method 1 :

Write the equation in slope intercept form.

y - 8  =  (-1/2)(x + 4)

Distributive property. 

y - 8  =  -x/2 - 2

Add 8 to each side.

y  =  -x/2 + 6

The above line is in slope intercept form. 

Comparing y = mx + b and y = -x/2 + 6, 

slope (m)  =  -1/2

Method 2 :

Write the equation in general form.

y - 8  =  (-1/2)(x + 4)

Distributive property. 

y - 8  =  -x/2 - 2

Add 2 to each side. 

y - 6  =  -x/2

Multiply each side by 2.

2(y - 6)  =  -x

Distributive property. 

2y - 12  =  -x

Add x to each side. 

x + 2y - 12  =  0

slope (m)  =  - coefficient of x / coefficient of y

slope (m)  =  - 1/2

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