Standard Form for the Equation of a Line
When we wrote the equation of a line in pointslope form, the equations
looked different depending on the point we chose. To show that the
equations are equivalent, rewrite each equation in standard form.
Definition â€”
Standard Form for the Equation of a Line
The standard form for the equation of a line is
Ax + By = C Where A, B, and C are real numbers and A and B are not both zero.
To write the equation of a line in standard form, move the terms with
variables to the left side and the constant term to the right side of the
equation.
Example 1
We can find the equation of the line that passes through
the points (2, 7) and (6, 3).
We may use the point (6, 3) to obtain:
If we instead use (2, 7), we obtain:
a. Write
in standard form.
b. Write
in standard form.
c. What conclusion can you draw?
Solution
In each case, we want the xterm and yterm on the left side of the
equation. We want the constant term on the right side of the equation.
a. 
To clear the fraction, multiply both
sides by 2.

2 Â· (y
 3) 


Simplify.
Distribute 1.
Add x to both sides.
Add 6 to both sides.

2y  6
2y  6
x + 2y  6
x + 2y 
= 1(x  6)
= x + 6
= 6
= 12 
b. 
To clear the fraction, multiply both
sides by 2. 
2 Â·
(y  7) 


Simplify.
Distribute 1.
Add x to both sides.
Add 14 to both sides. 
2y  14
2y  14
x + 2y  14
x + 2y 
= 1(x + 2)
= x  2
= 2
= 12 
c. In standard form, each equation is x + 2y = 12.
No matter which point we choose, we get the same result in standard form.
Note:
Ax + By = C
x + 2y = 12
Here, A is 1, B is 2, and C is 12.
