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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