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We will learn about one final form that an equation can take--general linear form. Equations in general linear form look like this:

Ax + By = C |

where

General linear form is not the most useful form to use when writing an equation from a graph. However, the form highlights certain abstract properties of linear equations, and you may be asked to put other linear equations into this form.

To write an equation in general linear form, given a graph of the equation, first find the *x*-intercept and the *y*-intercept -- these will be of the form (*a*, 0) and (0, *b*). Then one way to write the general linear form of the equation is

bx + ay = ab |

This equation is linear and the two intercept points satisfy it, therefore it represents the line. Finally, one should try to multiply or divide both sides of the equation by a number to make the coefficients as simple as possible. For instance, if

Another way to describe the same simplification procedure is that if (*a*, 0) and (0, *b*) are the *x*- and *y*- intercepts, respectively, and *a* and *b* are integers, then*C* = the least common multiple of *a* and *b**A* = *B* =

and *Ax* + *By* = *C* is an equation of the line.

If *a* or *b* is negative, take the positive least common multiple; i.e., the least common multiple of | *a*| and | *b*|. *A* or *B* will be negative, since we will be dividing a positive number by a negative number.

*Example 1*: Write an equation of the following line in general linear form:

Graph of a Line

Therefore, the equation of this line is 3

Check: 3(4) + 4(0) = 12 ? Yes.

3(0) + 4(3) = 12 ? Yes.

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