SLOPE OF THE LINE

About "Slope of the line"

Slope of the line :

It is the change in y for a unit change in x along the line and usually denoted by the letter "m"

Slope is sometimes referred to as "Rise over run"

Because the fraction consists of the "rise" (the change in y, going up or down) divided by the "run" (the change in x, going from left to the right).

The figure given below illustrates this. 

From the above figure, the slope of the straight line joining the points A (x₁, y₁) and B (x₂, y₂) is   

That is, 

If the equation of a straight line given in general form

ax + by + c  =  0,   

then, the formula to find slope of the line is 

Let θ be the angle between the straight line "l" and the positive side of x - axis.

The figure given below illustrates this. 

Then, the formula to find slope of the line is

m  =  tan θ

In the general form of equation of a straight line

ax + by + c  =  0,

(i)  if "x" term is missing, then the line will be parallel to x - axis and its slope will be zero.

We know that slope  =  change in y / change in x 

In the above figure, the value of "y" is fixed and that is "k"

So, there is no change in "y" and change in y = 0 

Slope  =  0 / change in x  

Slope  =  0 

(i)  if "y" term is missing, then the line will be parallel to y - axis and its slope will be undefined. 

We know that slope  =  change in y / change in x 

In the above figure, the value of "x" is fixed and that is "c"

So, there is no change in "x" and change in x = 0 

Slope  =  change in y / 0   

Slope  =  Undefined 

Slope of the coordinate axes "x" and "y". 

(i)  Slope of "x" axis zero.

(ii)  Slope of "y" axis undefined. 

Slope of the line - Positive or Negative or Zero or Undefined

When we look at a straight line visually, we can come to know its slope easily.  

To know the sign of slope of a straight line, always we have to look at the straight line from left to right. 

The figures given below illustrate this. 

Slope of the line - Practice problems

Problem 1 :

Find the angle of inclination of the straight line whose slope is           1/√3

Solution :

Let θ be the angle of inclination of the line. 

Then, slope of the line,  m  = tan θ

Given : Slope = 1/√3

So, we have 

tan θ  =  1/√3

θ  =  30°

Hence, the angle of inclination is 30° 

Problem 2 :

Find the slope of the straight line passing through the points (3, -2) and (-1, 4). 

Solution :

Let (x₁, y₁)  =  (3, -2) and (x₂, y₂)   =  (-1, 4)

Then, the formula to find the slope, 

m  =  (y₂ - y₁) / (x₂ - x₁)

Plug (x₁, y₁)  =  (3, -2) and (x₂, y₂)   =  (-1, 4)

m  =  (4 + 2) / (-1 - 3)

m  =  - 6 / 4

m  =  - 3 / 2

Hence, the slope is -3/2

Problem 3 :

Using the concept of slope, show that the points A(5, -2), B(4, -1) and C(1, 2) are collinear.

Solution :

Slope of the line joining (x₁, y₁) and (x₂, y₂) is, 

m  =  (y₂ - y₁) / (x₂ - x₁)

Using the above formula,

Slope of the line AB joining the points A (5, - 2) and B (4- 1) is 

 =  (-1 + 2) / (4 - 5)

=  - 1

Slope of the line BC joining the points B (4- 1) and C (1, 2) is 

 =  (2 + 1) / (1 - 4)

=  - 1

Thus, slope of AB = slope of BC.

Also, B is the common point.

Hence, the points A , B and C are collinear.

Problem 4 :

Find the slope of the straight line 3x - 2y + 7 = 0. 

Solution :

When the general form of equation of a straight line is given, the formula to find slope is

m  =  -  coefficient of x / coefficient of y 

In the given line 3x - 2y + 7 = 0,

coefficient of x  = 3 and coefficient of y  =  - 2

Slope,  m  =  (-3) / (-2)  =  3/2 

Hence, slope of the given line is 3/2. 

Problem 5 :

If the straight line 5x + ky - 1 = 0 has the slope 5, find the value of "k"  

Solution :

When the general form of equation of a straight line is given, the formula to find slope is

m  =  -  coefficient of x / coefficient of y 

In the given line 3x - 2y + 7 = 0,

coefficient of x  = 3 and coefficient of y  =  k

Slope,  m  =  -5 / k

Given : Slope  =  5

So, we have  5  =  -5/k

5k  =  -5

k  =  -1

Hence, the value of "k" is  -1. 

After having gone through the stuff given above, we hope that the students would have understood "Slope of a line". 

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