FINDING THE SLOPE OF A LINE FROM AN EQUATION

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.

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.

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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