General form for the equation of a line: 0 = Ax + By + C
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The formula 0 = Ax + By + C is said to be the 'general form' for the equation of a line. A, B, and C are three real numbers. Once these are given, the values for x and y that make the statement true express a set, or locus, of (x, y) points which form a certain line.
At first it is difficult to imagine the slant and position of the line just by looking at the general form. However, we can change the general form into slope-intercept form and then get a good idea of what we are looking at. Here's the algebra:
|0 = Ax + By + C||Starting equation|
|0 - By = Ax + By -By + C||Subtract By from each side|
|-By = Ax + C||Left side: 0 - By = -By
Right side: By - By = 0
|-By/-B = Ax/-B + C/-B||Divide each side by -B|
|y = (-A/B)x + (-C/B)||Left side: -By/-B = y
Right side: Ax/-B = (-A/B)x
Right side: C/-B = (-C/B)
|y = mx + b||Where: m = (-A/B)
Where: b = (-C/B)
As the previous algebra has shown, if we start with this general form:
0 = Ax + By + C
Then the slope of the line is:
slope = m = -A/B
And the y-intercept is:
y-intercept = b = -C/B
Let's start with this general form:
0 = Ax + By + C
A = 2, B = 3, C = 4
0 = 2x + 3y + 4
slope = m = -A/B = -2/3 = -0.67 (approximately)
y-intercept = b = -C/B = -4/3 = -1.33 (approximately)
Our line in slope-intercept form:
y = mx + b
y = -0.67x + (-1.33) or y = -0.67x - 1.33
Below is a graph that presents a line as it is given by the general form equation. You can control the example values for A, B, and C by clicking their appropriate '+' and '-' buttons. The current general form equation is shown in the upper left corner of the graph.
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