Various Forms of the Equation of a Line

Introduction to various forms of the equation of a line:

Definitions

The standard form of the equation line is defines as

Ax+By =C

Where A,B and C are represented as real numbers.
The point slope form of the equation of a line is represented given below
The equation of a line with slope represents m and passing through the point (x1,y1) is given by
y-y1 = m (x=x1)

Where m is the slope and x1, y1 is the point given
The slope intercept form of the equation of a line:
The equation of a line with slope m and y intercept (0,b) is given by
Y= mx+b

Where m is the slope and (0,b)is the y intercept.
If an equation represents a straight line is referred as the equation of the line. The following conditions are used to solve the lne equation.
when the slope and  y intercept is given
when the slope and a point is given.
when two points given for the line

Practice Problems for Various Forms of the Equation of a Line:

Solving the Equation of a Line when slope and y intercept is given:

The slope-intercept form of a line is y= mx + b,
where m is the slope

b is the y-intercept.

Example

Find the equation of the line with slope -4 and y-intercept 6.

Solution:                m = -4  and  b = 6

The general equation is y = mx + b
Substitute the values into equation, we get
y = -4x + 6

Hence, the equation of the line is y = -4x + 6

Solving  the Equation of a Line if slope and a point is given:

various forms of the equation of a line - Example: 1

Find the equation of a line in slope intercept form, if the slope is  -3 and passes through (5, 8)

Step 1: Use the slope-intercept form of a line: y = mx + b

Given m = -3     Hence y = - 3x + b

Step 2 : Substitute values into equation:

The y-intercept was not given. However, we are given the point, (5, 8). Thus x = 5 and y = 8
Substitute the values to find the y-intercept b
y  =  mx + b
8  = -3(5) + b
8  = -15 + b
23 = b
Step 3: Solution

apply  the value of b, we get  y = -3x + 23

Hence, the equation of the line is y = -3x + 23

Solving the Equation of a Line passing through two given points:

Example:

Find out the line that passes through (3, 5) and (7, 25)

Steps:

1)      Use the two points to find the slope using slope formula

2)      Use the slope and either one of the points to find the value the y-intercept.

Step 1:                       Slope          =   (y2– y1) / (x2 – x1)

=   (25– 5) / (7 – 3)

= 20/4 = 5

Step 2 :  Let’s choose the point (3,5)
y   =  mx + b
5  =  5(3) + b
5  = 15 + b
5 – 15 = b
-10 = b

Substituting the values in the general equation, we get

y = 5x -10

Hence the equation of the line is y = 5x -10.


Example for Various Forms of the Equation of a Line:

Example Problem 1

solve the standard equation of the line that passes through the points
( -2, 5 ) and ( 3, 8 ).

Solution:

The first step is to determine the slope of the line. According to determine the slope of the line, we must use the formula
m = ( y2 - y1 ) / ( x2 - x1 ). This gives us,

m = ( y2 - y1 ) / ( x2 - x1 )
m = ( 8 - 5 ) / ( 3 - ( -2 ) )
m = ( 8 - 5 ) / ( 3 + 2 )
m = 3/5

Therefore, the slope of the line is equal to 3/5.

by applying the point-slope formula. To do so, we must choose one of the points, ( x1, y1 ), and insert it and the slope into the formula which will give us,

y - y1 = m ( x - x1 )
y - 5 = ( 3/5 )( x - ( -2 ) )
y - 5 = ( 3/5 )( x + 2 )

Finally, now that we have the equation of the line in
point-slope form, we will want to convert the equation into
slope-intercept form. ( This will allow us to determine the
y-intercept directly from the formula. )

y - 5 = ( 3/5 )( x + 2 )
y - 5 = ( 3/5 )x +6/5
y = ( 3/5 )x6/5 + 5
y = ( 3/5 )x6/5 + 25/5
y = ( 3/5 )x +31/5

`(3/5)` x-y+`31/5` =0 is the equation of the line is standard form.