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So I was tutoring an Algebra 1 student yesterday and we were reviewing the three forms in which one can write the equation for a line: point-slope form, slope-intercept form and standard form. I told him to write an equation in standard form and he wrote:

$\frac x5 + \frac y4 = 1$ and I told him to rewrite it without any fractions as $4x +5y = 20$

Are both forms correct? I understand that the latter form may be more common but what I really want to know is if it is acceptable to use the former. Thanks all!

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I agree that the latter is more common, but the former is indeed acceptable: each describes precisely the same line, after all!

However, I think it's perfectly reasonable to suggest that, when presenting the equation of a line in "standard form," to strive for integer coefficients.

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The first form is usually called intercept form. That's because if you can write the equation in the form $\frac{x}{a} + \frac{y}{b} = 1$, then the $x$-intercept is $a$ and the $y$-intercept is $b$.

General form isn't unique, but intercept form is a specific variety of the general form in which the constant term is $1$ and is isolated on one side, and the coefficients of the variables are written as fractions with numerator $1$.

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  • $\begingroup$ I did not purposely choose 1, it was just a random constant I chose for this example. Your point is a good one, however. $\endgroup$ – user3175426 Feb 13 '14 at 18:35
  • $\begingroup$ For graphing, this is definitely the best way to go. Just move terms around so that the constant term is by itself on one side, divide by that term to get the desired form, read off the intercepts, and draw the line through them. How much easier could it get? Caveat: vertical lines and lines through the origin can't be represented this way. Those are easy to recognize in other forms, though. $\endgroup$ – MPW Feb 16 '14 at 0:25
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I have been taught that the standard form is: $$Ax + By + C = 0$$ Where A, B, and C are all integers, and A must be positive. So the latter part is correct. For example, if I have a "messy" equation in slope-intercept form, $$y = x/2 + 3/4$$ I would have to multiply both sides by 4, giving me the equation: $$4y = 2x + 3$$ Because the coefficient of the x term has to be positive, I will move 4y to the right, which gives: $$2x - 4y + 3 = 0$$ I hope this answer helped!

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