The general notation for a standard form equation is Ax + By = C, where A, B, and C are coefficients, and the x and y are the same variables we've been looking at but in a different position from what we recognize.
To express in standard form, you simply just rearrange the y = mx + b form such that you have x and y on the same side, equal to a number. Let's look at some examples:
Given that y = 3 x+ 5, standard form of this is 3x - y = (-5).
Given y = (1/2)x -15, standard form of this is (1/2)x - y = 15... also, if you don't want to have any fractions in your answer, you can multiply everything by the number in the denominator, such that we now get x - 2y = 30. Both expressions mean the same thing and will produce the same line. (In fact, convince yourself that no matter what you do to the equation, so long as you do it to both sides, the line is the same. eg. Multiply it all by 100, you get 100x-200y=30000... looks different, but it's not! Reduce it down and see for yourself!)
For graphing standard form equations, you still might want to go from standard form to the mx+b form, for which you may need to do a bit more math, but it's still quite straight forward.
Given 5x - 15y = 10, you just have to rearrange things to get y by itself on one side:
(-15y) = (-5x) + 10
y = (1/3)x - (2/3)...
and then you can see it is a line with slope 1/3 and y-intercept (-2/3).
Both types of equations mean the same thing. They are just expressed differently, and y=mx+b gives immediate information about the line without having to do a lot of work. However, you should be able to use both forms interchangeably. Convince yourself that graphing standard form equations will give you the same line as graphing y=mx+b equations. They just look different because the numbers are rearranged. This should be obvious because if you start with a standard form equation, and convert it to y=mx+b and graph it, you have only rearranged things not added or removed anything. You do not have a new line.
Also, from these equations, you should be able to tell that whenever you have an equation with 2 variables (x and y), and there aren't any exponents on either term, then you are dealing with a straight line. So while an equation in standard form may not immediately look like a straight line equation to you until it looks like y = mx + b, because it has an x and a y in it (without an exponent... exponents make the graph do cool things later), it is automatically a straight line.
(check this post for some additional pointers)
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