HOW TO FIND THE EQUATION OF A STRAIGHT LINE

General form of equation of a straight line

ax + by + c = 0

Standard form of equation of a straight line

ax + by = c

Different Forms of Equations of Straight Line

Apart from general form and standard form of equation of a straight line, we have some other different forms of equation of a straight line which are much important to find the equation of a straight line in different situation. 

The different forms of equation of straight line are

1. Slope - Intercept form :

y = mx + b

Slope =  m

y-intercept  =  b

2. Point - Slope form :

y - y₁ = m(x - x₁)

Point  =  (x₁ , y₁)

Slope  =  m

3. Two points form :

First point = (x₁ , x₁)

Second point = (y₂ , y₂)

4. Intercepts form :

x-intercept = a

y-intercept = b

Slope of a Straight Line

Stuff 1 :

If the equation of a straight line is in the form of

ax + by + c = 0, 

then, the slope is

= -a/b

That is,

slope = - coefficient of x/coefficient of y

Stuff 2 :

If two lines are parallel, the slopes of the two lines would be equal.

That is,                                

m₁ = m₂

Here m₁ and m₂ are the slopes of the two straight lines.

Stuff 3 :

If two lines are perpendicular, the product of the slopes of two lines would be equal to -1.

That is,

m₁ x m₂ = -1

Here m₁ and m₂ are the slopes of the two straight lines.

Stuff 4 :

If the coefficients of x and y of two straight lines are equal, then the lines would be parallel.

Example :

2x + 3y - 5 = 0  and  2x + 3y + 1 = 0

In the above example,

Coefficients of x in both lines are same and it is 2.

Coefficients of y in both lines are same and it is 3.

So the two straight lines

2x + 3y - 5 = 0

2x + 3y + 1 = 0

Moreover, slope of the line 2x + 3y - 5 = 0 is -2/3.

Slope of the line 2x + 3y + 1 = 0 is -2/3.

How to find the equation of a straight line ?

Step 1 :

First we have to go through the question carefully and understand the information given in the question.

Step 2 :

Based on the information given in the question, we have to use one of the four different forms of equation of straight line. 

(i) Slope intercept form

(ii) Point slope form

(iii) Two points form

(iv) Intercepts form

Step 3 :

For example, if the information given in the question related to 'Slope intercept form', we have to use

y = mx + b

to get the equation of a straight line.

Step 4 :

If the question demands the answer in general form, our answer must be in the form

ax + by + c = 0

If the question demands the answer in standard form, our answer must be in the form

ax + by = c

In case, the question does not demand the answer in a particular form, we can give the answer in the form we get.

Example

Find the equation of a line passing through the point(5, -4) and parallel to the line 4x + 7y + 5 = 0.

Step 1 :

The required line is passing through (5, -4)

The required line is parallel to the line 4x + 7y + 5 = 0.

Step 2 :

Slope of the given line 4x + 7y + 5 = 0 is -4/7.

Required line is parallel to 4x + 7y + 5 = 0.

So the slope of the required line is -4/7.

Step 3 :

From step 1 and step 2, it is very clear that the required line is passing through (5, -4) and having slope -4/7.

Step 4 :

Since we have a point and slope, we have to use the form

y - y₁ = m(x - x₁)

Here  (x₁, y₁) = (5, -4) & m = -4/7.

Step 5 :

Substitute x₁ = 5, y₁ = -4 and m = -4/7. 

y - (-4) = -4/7(x - 5)

7(y + 4) = -4(x - 5)

7y + 28 = -4x + 20

4x + 7y + 8 = 0

So, the equation of the required line is

4x + 7y + 8 = 0

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