DIFFERENT FORMS OF EQUATION OF A STRAIGHT LINE

Example 1 :

A straight line with the gradient of -2 passes through the point (4, -1). Find the equation of the line in slope intercept form.

Solution :

Slope of the required line = -2

The required line is passing through the point (4, -1).

Equation of the line :

(y - y₁) = m(x - x₁)

(y - (-1)) = -2(x - 4)

y + 1 = -2(x - 4)

Distributing -2, we get

y + 1 = -2x + 8

Subtract 1 on both sides, we get

y = -2x + 8 - 1

y = -2x + 7

Example 2 :

A straight line passes through the points A(1, 0) and B(3, 6). Find the gradient of the line and find equation in standard form.

Solution :

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope of the line joining the points A(1, 0) and B(3, 6). 

x₁ = 1, x₂ = 3, y₁ = 0 and y₂ = 6

m = (6 - 0)/(3 - 1)

m = 6/2

m = 3

Equation of the line in slope intercept form :

y = mx + b

y = 3x + b ----(1)

Applying the point (1, 0), we get

0 = 3(1) + b

b = -3

Applying the value of b in (1), we get

y = 3x - 3 (Slope intercept form)

Converting into standard form, we get

3x - y - 3 = 0

Example 3 :

4x + 3y = 6 is a straight line. Find the gradient and y-intercept of the line.

Solution :

4x + 3y = 6

Subtracting 4x on both sides.

3y = -4x + 6

Dividing by 3 on both sides.

y = (-4/3)x + (6/3)

y = (-4/3)x + 2

Slope (m) = -4/3 and y-intercept = 2.

Example 4 :

Write the equation of the line which is parallel to the y-axis and passes through (1, 4)

Solution :

Equation of the line parallel to y-axis is y = 4.

Example 5 :

Write the equation of the line which is parallel to the x-axis and passes through (3, -5) ?

Solution :

Equation of the line parallel to x-axis is y = 3.

Example 6 :

Find the slope and y-intercept of the lines given below.

(i) 4x + y = 9

(ii)  3x - 2y = 4

Solution : 

Converting the given standard form to slope intercept form, we can find slope and y-intercepts.

(i) 4x + y = 9

Subtract 4x on both sides.

y = -4x + 9

Comparing it into y = mx + b, we get

Slope = -4 and y-intercept = 9

(ii)  3x - 2y = 4

Add 2y on both sides.

3x = 2y + 4

Subtract 4 on both sides.

3x - 4 = 2y

Divide by 2 on both sides.

y = (3/2)x - (4/2)

y = (3/2)x - 2

Slope (m) = 3/2 and y-intercept = -2.

Example 7 :

Graphically, the pair of equations 6𝑥– 3y + 10 = 0 , 2𝑥– y + 9 = 0 are represented by two lines that are

(a)  Intersecting (b)  Parallel (c)  coincident (d)  either Intersecting or Parallel

Solution :

6𝑥 – 3y + 10 = 0

2𝑥 – y + 9 = 0

To find whether these lines are intersection, parallel or coincident, we have to convert each line in the slope intercept form and find the slope and y-intercepts.

Since the lines are having same slope and different y-intercepts, these two are parallel.

Example 8 :

What is the value of p if, if the following pair of the equations 2𝑥 + 3y – 5 = 0, p𝑥 – 6y – 8 = 0 has a unique solution.

(a) p ≠ -4  (b) p = -4   (c) p = 4   (d) p = -1

Solution :

When the lines are intersection they should not have same slope.

-2/3 ≠ p/6

-2(6) ≠ 3p

p ≠ -12/3

p ≠ -4

So, option a is correct.

Example 9 :

The pair of the equations 𝑥 + 2y + 5 = 0, –3𝑥 – 6y + 1 = 0 has

(a) unique solution     (b) exactly two solutions

(c) infinitely many solutions    (d) no solution

Solution :

Since the slopes are equal and y-intercepts are not equal, then the lines will be parallel and there is no solution.

Example 10 :

If the lines 3𝑥 + 2ky – 2 = 0 and 2𝑥 + 5y + 1 = 0 are parallel, then what is the value of k?

Solution :

Since the lines are paralle, their slopes will be equal.

m₁ = m₂

-3/2k = -2/5

-15 = -4k

k = 15/4

So, the value of k is 15/4.