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

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 1 :

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

Solution :

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

Example 2 :

Find the value of p if the lines 5x - 3y + 2 = 0 and 6x - py + 7 = 0 are perpendicular to other. Hence find the equation of a line passing through (-2, -1) and parallel to 6x - py + 7 = 0

Solution :

When two lines are parallel, then their slopes of will be equal.

5/3 = 6/p

p = 6(3)/5

p = 18/5

So, the required value of p is 18/5.

Equation of the line which passes through the point (-2, -1) and has the slope 5/3.

(y - y₁) = m(x - x₁)

y + 1 = 5/3(x + 2)

3(y + 1) = 5(x + 2)

3y + 3 = 5x + 10

5x - 3y = 3 - 10

5x - 3y = -7

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