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.
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.
Changing to standard form
Write the equation
in standard form using only integers and a positive
coefficient for x.
Use the properties of equality to get the equation in the form Ax + By = C:
from each side.
||Multiply each side by 4 to
get integral coefficients.
|-2x + 4y
|2x - 4y
||Multiply by -1 to make the
coefficient of x positive.
To find the slope and y-intercept of a line written in standard form, we convert
the equation to slope-intercept form.
Changing to slope-intercept form
Find the slope and y-intercept of the line 3x - 2y = 5.
Solve for y to get slope-intercept form:
|3x - 2y
||= -3x + 5
||Subtract 3x from each side.
||Divide each side by -2.
The slope is
, and the y-intercept is
Solve Ax + By = C for y, to
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.