FINDING THE SLOPE OF A LINE FROM AN EQUATION

The easiest way to find the slope of a line from its equation is writing the given linear equation in slope-intercept form, that is

y = mx + b

where m is the slope and b is the y-intercept.

Once you write the equation equation in slope-intercept form, easily we can get the slope, that is the coefficient of x (m) on the right side of the equation. 

Slope of vertical and horizontal lines :

The general equation of a vertical line :

x = a

In general, slope of a line :

= change in y-values/change in x-values

In a vertical line, no change in x values, that is,  x-value is always a constant.

So, the slope of a vertical line is

= change in y-values/0

= undefined

Slope of a vertical line is undefined.

The general equation of an horizontal line :

y = b

In an horizontal line, no change in y values, that is y-value is always a constant.

So, the slope of an horizontal line is

= 0/change in x-values

= 0

Slope of an horizontal line is zero.

Example 1 :

Find the slopes of the straight lines :

(i) 3x + 4y – 6 = 0

(ii) y = 7x + 6

(iii) 4x = 5y + 3

Solution :

(i) :

3x + 4y – 6 = 0

Write the given equation in slope-intercept form.

4y = -3x + 6

y = (-3/4)x + 3/2

The above equation is in slope-intercept form.

Comparing 

y = mx + b

and

y = (-3/4)x + 3/2,

we get

m = -3/4

slope = -3/4

(ii) :

y = 7x + 6

The given equation is already in slope-intercept form.

Comparing 

y = mx + b

and

y = 7x + 6,

we get

m = 7

slope = 7

(iii) :

4x = 5y + 3

Write the given equation in slope-intercept form.

4x - 3 = 5y

(4/5)x - 3/5 = y

or

y = (4/5)x - 3/5

Comparing 

y = mx + b

and

y = (4/5)x - 3/5,

we get

m = 4/5

slope = 4/5

Example 2 :

Show that the following straight lines are parallel.

x + 2y + 1 = 0

3x + 6y + 2 = 0

Solution :

If two lines are parallel, then their slopes are equal.

Write each of the given two equations in slope intercept form.

x + 2y + 1 = 0

2y = -x - 1

y = (-1/2)x - 1/2

slope = -1/2

3x + 6y + 2 = 0

6y = -3x - 2

y = (-1/2) - 1/3

slope = -1/2

Since, the slopes are equal, the given two lines are parallel.

Example 3 :

Show that the following two straight lines are perpendicular.

3x – 5y + 7 = 0

15x + 9y + 4 = 0

Solution :

If two lines are perpendicular, then the product of the slopes is equal to -1.

Write each of the given two equations in slope intercept form.

3x - 5y + 7 = 0

-5y = -3x - 7

5y = 3x + 7

y = (3/5)x + 7/5

slope = 3/5

15x + 9y + 4 = 0

9y = -15x - 4

y = (-15/9)x - 4/9

y = (-5/3)x - 4/9

slope = -5/3

Product of the slopes :

= (3/5) ⋅ (-5/3)

= -1

Since, the product of the slopes is equal to -1, the given two lines are perpendicular.

Example 4 :

Find the slope of the straight line x = -4.

Solution :

x = -4 is the equation of a vertical line which is passing through the value -4 on the x-axis. 

Since the slope of a vertical line is undefined, slope of the line x = -4 is also undefined.

Example 5 :

Find the slope of the straight line y = 2.

Solution :

y = 2 is the equation of an horizontal line which is passing through the value 2 on the y-axis. 

Since the slope of an horizontal line is zero, slope of the line y = 2 is also zero.

Example 6 :

Find the slope of the straight line x = 0.

Solution :

x = 0 is the equation of y-axis.

y-axis is always a vertical line and its slope is undefined.

Therefore, slope of the line x = 0 is undefined.

Example 7 :

Find the slope of the straight line y = 0.

Solution :

y = 0 is the equation of x-axis.

x-axis is always an horizontal line and its slope is zero.

Therefore, slope of the line y = 0 is zero.

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