Linear equation

The equation representing a line is called linear equation. There are different methods of representing a line.  Slope intercept form is the form of line which is used to represent a line using  the slope and the intercept.
Standard form is a form which is used to represent a line using two intercepts.

Slope Intercept form

If m is the slope and c is the intercept of a line, then the slope intercept form is given by y= mx + c.

Standard form

The standard form of a line is given by Ax + By = C where A, B and C are constants.

Lets learn how to convert the slope intercept form of a line to standard form

Slope intercept form to standard form

When we take all the variables to one side and keep the constants on the other side, we get the standard form.
That is,
Given slope intercept form is y = mx + c
Subtract mx from both sides, we get
y - mx  = c
Rearranging we get -mx + y = c, which is the required equation.

If there is a fraction in the constants, multiply all the terms by the least common factor of the denominators and then take all variables to one side keeping the constants on the other.Lets consider a few examples.

Example 1: -
Convert the line y = 3x + 7 into standard form.
Solution: -
Given line is y = 3x + 7.  To convert into standard form, we have to take all variables to one side, keeping the constants on the other side.
For this lets subtract 3x from both sides,
   y   = 3x + 7
-3x      -3x      
y - 3x =       7
So the required equation is -3x + y = 7

Example 2: -
Convert the line y = -5x + 8 into standard form.
Solution: -
Given line is y = -5x + 8.  To convert into standard form, we have to take all variables to one side, keeping the constants on the other side.
For this lets subtract -5x from both sides.  That is add 5x on both sides,
   y   = -5x + 8
 5x        5x      
y + 5x =       8
So the required equation is 5x + y = 8

Example 3: -
Convert the line y = -3/2 x +8 into standard form.
Solution: -
Given line is y = -3/2 x  + 8 To convert into standard form, we have to take all variables to one side, keeping the constants on the other side.
Here we have a fraction in the equation. So we multiply the whole equation by the denominator 2.
We get
2y = -3x +16
Now we take all variables to one side keeping the constants to other. For this lets subtract -3x from both sides.  That is add 3x on both sides,
   2y     = -3x + 16
   3x          3x      
2y + 3x =          16
So the required equation is 3x + 2y = 16

Example 4: -
Convert the line y= -2/3 x  + 8/7 into standard form.
Solution: -
Given line is y= -2/3 x  + 8/7 To convert into standard form, we have to take all variables to one side, keeping the constants on the other side.
Here we have two fractions; 2/3 and 8/7 in the equation. The Least common multiple of 3 and 7 is 21. So we multiply the whole equation by the least common multiple 21.
We get
21y = -14x + 24
Now we take all variables to one side keeping the constants to other. For this lets subtract -14x from both sides.  That is add 14x on both sides,
   21y     = -14x + 24
   14x            14x      
21y +14x =           24
So the required equation is 14x + 21y = 24

Example 5: -
A line is passing through the point (2, 3) with slope 5.  Write the equation in standard form
Solution: -

Given the point through which the line is passing is (2, 3).
Slope of the line = 5.
First lets write the line using point slope form.
We know that when a point (x1, y1) and slope m is given, then the point slope form is
y - y1 = m (x - x1).
Substituting the slope and points we get
y - 3 = 5 (x - 2)
Expanding we get
y - 3 = 5x - 10
Now we to take all variables to one side, keeping the constants on the other side.
Adding 3 on both sides, we get
y - 3 = 5x - 10
   +3         + 3

y     =  5x  - 7
Now we subtract 5x from both sides
   y     = 5x - 7
   -5x            -5x      
y -5x =           -7
So the required equation is -5x + y = -7

Try Yourself: -

Write the equation in standard form

1. y = 4x + 3
2. y = -2/3 x + 5

Answers: -
1. -4x +  y  = 3
2. 2x + 3y = 15