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 A, B, and C are integers, is the x-intercept, and is the y-intercept.
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 a and b are fractions, one can multiply both sides by a common denominator to obtain integer coefficients. Once the coefficients are integers, one can divide by their greatest common divisor to simplify even further.
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
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:
The x-intercept is (4, 0) and the y-intercept is (0, 3). Thus, a = 4 and b = 3. The LCM of 4 and 3 is 12. Thus, C = 12
A = = = 3
B = = = 4
Therefore, the equation of this line is 3x + 4y = 12.
Check: 3(4) + 4(0) = 12 ? Yes.
3(0) + 4(3) = 12 ? Yes.
Example 2: Write an equation of the line which passes through (0, 8) and (- 6, 0).
C = the LCM of 8 and 6 = 24.
A = = - 4
B = = 3
Thus, an equation of the line is -4x + 3y = 24. If we wanted to write an equation with a positive value first, we could write 4x - 3y = - 24.
To graph an equation in general linear form, compute the x-intercept (a, 0) and the y-intercept (0, b): a = and b = . Then connect the intercepts with a straight line and extend the line on both sides.