Standard Form of a Line
If x students paid $5 each and y adults paid $7 each to attend a play for which the
ticket sales totaled $1900, then we can write the equation 5x + 7y = 1900. This
form of a linear equation is common in applications. It is called standard form.
Standard Form
The equation of a line in standard form is
Ax + By = C, where A, B, and C are real numbers with A and B not both zero.
The numbers A, B, and C in standard form can be any real numbers, but it is a
common practice to write standard form using only integers and a positive coefficient for x.
Example 1
Changing to standard form
Write the equation
in standard form using only integers and a positive
coefficient for x.
Solution
Use the properties of equality to get the equation in the form Ax + By = C:
y 

Original equation 


Subtract
from each side.



Multiply each side by 4 to
get integral coefficients. 
2x + 4y 
= 3 
Distributive property 
2x  4y 
= 3 
Multiply by 1 to make the
coefficient of x positive. 
To find the slope and yintercept of a line written in standard form, we convert
the equation to slopeintercept form.
Example 2
Changing to slopeintercept form
Find the slope and yintercept of the line 3x  2y = 5.
Solution
Solve for y to get slopeintercept form:
3x  2y 
= 5 
Original equation 
2y 
= 3x + 5 
Subtract 3x from each side. 
y 

Divide each side by 2. 
The slope is
, and the yintercept is
.
Helpful Hint
Solve Ax + By = C for y, to
get
So the slope of Ax 6+ By = C is
This fact can be used in
checking standard form. The
slope of 2x  4y = 3 in Example 2 is
, which is the
slope of the original equation.
