Slope of a line

We all are very much familiar with geometry, to be precise two dimensional one – mainly it is a combination of algebra and geometry. Rene Descartes was the first mathematician to study this aspect. We all are familiar with plotting of points in a plane, distance between two points, section formulae and various other basic concepts of coordinate geometry. Here we shall look at various concepts related to the slope of a line. Widely used formula for Slope of a line is

tan θ = y2–y1/x2–x1

Slope of a line

The smallest angle made by a line with the positive direction of the X-axis measured in counter clockwise sense is known as inclination of the line. We denote inclination of the line by θ.

Now if the inclination of a line is θ then tan θ is known as the slope of the line. It is denoted as m 

Mathematically m is expressed as:

m = tan θ

and tan θ = y2–y1/x2–x1 , where x and y are the coordinates.

One can say that two lines are parallel if and only if they have the same slope.

Before proceeding towards the various formulae of slope reader must be familiar with the below theorems 

• Theorem 1: Non-vertical lines have slopes m1 and m2 and are perpendicular to each other if and only if m1*m2 = -1
• Theorem 2: if θ is the acute angle between non-vertical lines have slopes m1 and m2 then 

Tan θ = │m1–m2/1+m1m2│

Equation of a line in standard form

If there is an equation in x and y which is satisfied by the coordinates of all points of a line and no other points then it is called the equation of the line.

• The Y coordinate of every point on the x-axis is 0 and this is true only for points on the X axis. Therefore, equation of X axis is y =0
•  Similarly, the equation of the y axis is x=0

The equation of any line parallel to the Y-axis is of the type x =k , (k is constant) and the equation of any line parallel to the X-axis is of the type y =k

Point – slope form

Here we are not concerned about the proof of the point – slope form as we are very interested in the result that we get for such form of the slope, so the equation of the line having slope m and which passes through the point A(x1 , y1) is given as :

y- y1=m( x- x1)

Reader’s must take note that if the line passes through the origin (0,0) and has a slope m , then the line equation reduces to :

Y – 0 = m ( x -0 )

Y = mx 

Example:

Find the equation of a line passing through the point A(2,1) and having slope -3

Solution: we know that the equation of a line passing through point A and having slope is given as 

y- y1=m( x- x1)

Therefore, Y -1 = -3 ( x- 2)

y-1 = -3x +6

3x +y -7 = 0

This is the required equation.

Slope – intercept form 

The equation of line having slope m and which makes intercept c on the y-axis is y = mx+c

Example: obtain the equation of line having slope 3 and which makes intercept 4 on the y – axis.

Solution:  we will make use of the above formula here i.e., y = mx +c 

Y = 3x + 4 

Two-point form

We know that slope of a line = y-y1x- x1

So, if we just adopt the above formula for the points mentioned above, we can get the equation of the line. 

The equation of the line which passes through points A(x1, y1) and B(x2 , y2)  is x- x1 x2–x1= y-y1y2-y1

Example: Obtain the equation of the line passing through points A(2,1) and B(1,2).

Solution: the equation of the line which passes through points A(x1 , y1) and B(x2 , y2) is 

 x- x1 x2–x1= y-y1y2-y1

Therefore, as per the above question we get, 

x-2-1= y-11

x -2 = -y +1

x + y -3 =0

Types of slope

• Positive slope
• Negative slope
• Undefined slope
• Zero slope

Positive slope

As the name suggests a positive slope means positive growth i.e., when x increases then y also increases. And when y decreases then x also decreases. Graphically that would mean rise in the line moving left to right.

Negative slope

Here too the name speaks for itself, here the relationship between the two variables that is, x and y are basically inversely related. If x increases then y decreases and if y increases then x decreases. Graphically such a slope means that when we move from left to right the line falls. 

Zero slope

Zero slope means that y remains constant no matter what the value of x. graphically the line of such a slope is flat.

Undefined slope

We can understand it in this way that This is just the reverse of the zero slope that x remains constant no matter what the value of y is. The lines rise straight up or fall down straight.

Conclusion 

The smallest angle made by a line with the positive direction of the X-axis measured in counter clockwise sense is known as inclination of the line. We denote inclination of the line by θ.Now if the inclination of a line is θ then tan θ is known as the slope of the line. It is denoted as m. There are 4 types of slopes: positive slope, negative slope, undefined slope and zero slope.