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When a linear equation is written in standard form $Ax+By=C$ can $C$ be negative?

ie. is it true to say that $-4x+3y=5$ is written in standard form whereas $4x-3y=-5$ is not?

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2 Answers

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Usually such a requirement is not implied by saying that the equation is in standard/canonical/whatever form.

If, for some particular application, it is easier for you to handle equations with non-negative $C$, then you're of course free to set up such a requirement yourself. (If you get an equation that doesn't satisfy it, you can just multiply both sides by $-1$).

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It really depends on the context. For solving linear equations by substitution, the standard form of $-4x+3y=5$ would be multiplying by $\frac{1}{4}$ to obtain $x=\frac{3y-5}{4}$ for substitution. So the normal form could be $$ -4x+3y-5=0, $$

or $$ \; x-\frac{3}{4}y+\frac{5}{4}=0. $$ The last normal form automatically determines the sign of the constant term.

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