How to find the Equation of a Straight Line?

Geometry is a field of mathematics that deals with the size, position, shapes, angles, and dimensions of various things. 2D geometry includes shapes like squares, circles, triangles, hexagons, etc .2D shapes only have 2 dimensions. 3D geometry includes shapes like cubes, cylinders, cones, etc.3D shapes have 3 dimensions. But all these shapes are made with lines. But what exactly is a straight line.

A straight line is a 1 Dimensional quantity. It is defined as the shortest distance between two points. Multiple lines can be drawn between any two given points but only one of them is straight. That line is the shortest distance between both the points and it is called a Straight Line.

Various forms of equations of Straight lines

General form of a straight line 

ax + by + c =0

Here,

a, b, c are constants 

x, y are variables

In this form, the slope of the line is given by m = -a/b

Example: 2x + 3y + 5 =0

Question: What will be the slope of the line 5x + 3y -5 = 0.

Answer: 

The slope of the line ax + by + c=0 is -a/b

comparing with given line 5x + 3y – 5 = 0

In this form, the slope of the line is given by m = -a/b

Hence the slope of the given line is -5/3.

Slope intercept form

The slope-intercept form of a straight line is

y = mx + c

 Here m is the slope of the line and c is the y-intercept ( the value of y when x = 0)

Example: y = 2x + 5

Question: What is the slope-intercept of the line  5x – 3y +6 = 0?

Answer:

The slope intercept form of a straight line is y = mx + c

 m is the slope, slope of a straight line is 5/3 

c is the slope, substitute x=0

=> 5(0) – 3y = -6

=> y=2

Therefore c = 2

hence the equation of the line is y = 5x/3 + 2  

                                                     y = (5x + 6)/3

                                                     3y = 5x + 6

                                                => 5x – 3y + 6

Slope point form

The slope point form of a straight line is

y-y₁ = m(x-x₁)

here x and y are variables 

m is the slope 

x_1, , y₁ is a point on the straight line. 

Example: y – 5 = 2 ( x – 3)

Question: Check if the lines 3x + 8y + 5 = 0 and 8x – 3y + 1 = 0 are perpendicular or not.

Answer:

The slope of the first line 3x + 8y + 5 =0 is -a/b

=> m = -3/8

The slope of the second line 8x – 3y + 1 =0 is -a/b

=> m=8/3

the product of slope is -1 

So both the lines are perpendicular.

Question: What is the equation of the line passing through (2,4) with a slope of 6 .

Answer:

The slope point form of a straight line id y-y₁ = m(x-x₁)

Hence the equation is y – 4 = 6(x – 2)

                                          => y – 4 = 6x -12

                                          => 6x -y -8 = 0

Therefore the general form is 6x -y -8 = 0

Two-point form

The two-point form of a straight line is

(y-y₁)/(y₂-y₁₎= (x-x1)/(x₂-x₁)

here x and y are variables

(x₁, y₁) and (x₂, y₂₎ are two-point on the straight line. 

Example: (y – 5)/(3) =  ( x – 3)/(2)

Line in two point form

Question: What is the equation of the line passing through (2,4) and (3,8).

Answer: 

The Two point form of a straight line id (y-y₁)/(y₂-y₁) = (x-x1)/(x₂-x₁)

Hence the equation is (y – 4)(8-4) = (x – 2)(3-1)

                                         => (y – 4)/4 = 2x -2

                                         => 8x -y -4 = 0

Therefore the general form is 8x -y -4 = 0

Intercept form 

The intercept form of a straight light is x/a +y/b =1

Here x and y are variable 

a and b are intercepts of x and y respectively 

example : x/3 + y/4 =1

A Straight line in intercept form

Question: What is the equation of the line whose x and y-intercepts are 2 and 3

Answer: 

The intercept form of a straight light is x/a +y/b =1

Hence a=2 and  b=3

=> x/2 + y/3 =1

=> 3x + 2y = 6

Therefore the general form is 3x+2y -6= 0

If any two straight lines are equal the slopes of both lines are equal and if any two straight lines are equal the product of the slope is -1.

Sample Questions

Question 1: What is the equation of the line whose x and y-intercepts are 4 and 3?

Answer:

The intercept form of a straight light is x/a +y/b =1

Hence a=2 and  b=3

=> x/4 + y/3 =1

=> 3x + 4y = 12

Therefore the general form is 3x+4y -12= 0

Question 2: What is the equation of the line passing through (5,4) and (4,8).

Answer:

The Two point form of a straight line id (y-y₁)/(y₂-y₁) = (x-x₁)/(x₂-x₁)

Hence the equation is (y – 4)(8-4) = (x – 5)(5-4)

=> (y – 4)/4 = x -5

=> 4x -y -16 = 0

Therefore the general form is 4x -y -416= 0

Question 3: What is the equation of the line passing through (3,7) with a slope of 2.

Answer:

The slope point form of a straight line id y-y₁= m(x-x₁)

Hence the equation is y – 7 = 2(x – 3)

=> y – 7 = 2x – 6

=> 2x -y + 1 = 0

Therefore the general form is 2x -y +1 = 0

Question 4: What is the slope-intercept of the line  5x – 6y +6 =0.

Answer:

The slope intercept form of a straight line is y = mx + c

m is the slope, slope of a straight line is 5/6

c is the slope, substitute x=0

                       => 5(0) – 6y = -6

                       => y=1

Therefore c =1

hence the equation of the line is y = 5x/6 + 1

                                                    y = (5x + 6)

                                                    y = 5x + 6

                                               => 5x – y + 6

Therefore the general form is 5x – y + 6= 0

Question 5: What will be the slope of the line 2x + 3y -5 =0.

Answer:

The slope of the line ax + by + c=0 is -a/b

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

Question 6: What is the equation of the line whose x and y-intercepts are 5 and 3 also write the slope-intercept form.

Answer:

The intercept form of a straight light is x/a +y/b =1

Hence a=5 and  b=3

=> x/5 + y/3 =1

=> 3x + 5y = 15

Therefore the general form is 3x+5y – 15= 0

The slope of the line is -3/5

Hence the slope intercept form is y = mx +c

                                                      y = -3x/5 + 3

                                                     3y = -3x + 15

                                                     3x + 3y -15 = 0

                                                      x + y -5 =0

Question 7: What will be the slope of the line 3x + 3y -5 =0.

Answer: 

The slope of the line ax + by + c=0 is -a/b

Hence the slope of the given line is -3/3 = -1

Question 8: Check if the lines 8x + 9y + 5 = 0 and 9x – 8y + 1 = 0 are perpendicular or not.

Answer:

The slope of the first line 8x + 9y + 5 =0 is -a/b

 => m₁ = -8/9

 The slope of the second line 9x – 8y + 1 =0 is -a/b

=> m₂ = 9/8

the product of the slope of perpendicular line is  -1

here, m₁×m₂ = -1

So both the lines are perpendicular