How do you write ##y=2/3x+4## in standard form?

##2x-3y=-12##

The equation:

##y = 2/3x+4##

is in intercept form, which is a standard form of equation for a line.

Subtracting ##y## from both sides, we get:

##0 = 2/3x-y+4##

Subtracting ##4## from both sides we get:

##-4 = 2/3x-y##

Multiplying both sides by ##3## and transposing, we get:

##2x-3y = -12##

This is in what some authors call standard form:

##Ax+By=C##

with a preference that ##A## is a positive integer.

Note that:

  • The coefficients ##A##, ##B## and ##C## are not unique. You can multiply them all by the same non-zero factor to get another equation for the same line.
  • If the line is horizontal, then ##A=0## and cannot be a positive integer.
  • There may be additional “rules” to prefer ##B## to be an integer too, but that would not be possible in the case of lines with irrational slope.
  • Some authors do not specify that ##A## should be an integer.

Note that if you wanted the standard form of a polynomial, then the following form would be preferred:

##Ax+By+C = 0##

In our example, that would be ##2x-3y+12 = 0##

In the context of equations of lines, this is sometimes known as general form.

For standard polynomial form, integer coefficients with positive ##x## coefficient and no common factor are preferred if possible.