Standard Form of a LineDate: 04/23/2002 at 22:48:05 From: Tricia Swanson Subject: Standard Form What is standard form? What does Ax+by = c mean and what do the letters stand for? Date: 04/24/2002 at 23:05:27 From: Doctor Peterson Subject: Re: Standard Form Hi, Tricia. This is what is sometimes called the "standard form" for a line. (Some people think of slope-intercept as the standard form.) It is most commonly used when we write systems of linear equations. Although the slope-intercept form is the most familiar form, because it expresses y directly in terms of x, not all lines can be written that way, because vertical lines have no slope, and have to be written as x = k. But all lines can be written in standard form, because it is symmetrical (treating x and y in the same way). In particular, the line y = mx + b can be written as mx - y = -b which is in your "standard form" - the sum of a multiple of x and a multiple of y is equal to a constant. And the line x = k is already in "standard form," with A = 1, B = 0, and C = k. This form doesn't tell you what y is for a given x, because that can't be done for all lines. What it does is to say that any point (x,y) for which Ax + By has a particular value is on this line. So the value of standard form is that it can be applied to all lines, including vertical lines, which can't be written in slope-intercept form. And that, in turn, is because it is "unbiased": it treats x and y in the same way, rather than putting y in a special position. What's awkward about "standard form" is that the parameters A, B, and C do not stand for something obvious, as in the other forms. In fact, they are not fixed; if you double them all, you still have the same line. But, rewriting the standard form in slope-intercept form, Ax + By = C y = -A/B x + C/B and finding the x-intercept y=0 => x = C/A you can determine that the slope is m = -A/B and x- and y- intercepts are a = C/A b = C/B So the ratios of parameters in this form tell you about the line. Note that if A is zero, there is no x-intercept, and if B is zero, there is neither an x-intercept nor a slope. And that makes sense if you look at the lines. Notice also that the slope is independent of C; if you change C, you make a line parallel to the original line. Converting between standard form and slope-intercept form shows how they are related. This also serves to explain an important fact about the standard form, namely that although A, B, and C have no direct graphical interpretation, they are related to the visible concepts of slope and intercept. That mitigates the objection that the standard form is not helpful in graphing. I can add a couple of additional reasons why the standard form is worth using: 1. It fits well with a vector formulation of geometry that you may see in the future; using vectors, it can be written as a "dot product": (A,B).(x,y) = C 2. It relates the equation of a line to what is called "linear combination" in more advanced mathematics, namely the sum of multiples Ax + By. This has use far beyond graphing lines. But mostly it is important just because it is a "standard" that can be used for all lines, not a special form that makes it easier to visualize those lines for which it can be used, as with slope-intercept. You will find a variation on standard form, called General Form, in the Dr. Math FAQ (select Two Dimensions: Lines): Analytic Geometry Formulas http://mathforum.org/dr.math/faq/formulas/faq.analygeom_2.html - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 04/24/2002 at 23:09:15 From: Tricia Swanson Subject: Standard Form Thank you so much for the help! This site is wonderful! |
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