# Graphing: Standard Form of the Equation

Just a short explanation for what is meant by “standard form” of the equation of the line. We have been looking at line equations in the form of y=mx+b. However, you may be asked to express this in standard form, or as a standard form equation.  Graphing standard form equations will give you the exact same line as graphing something expressed as y=mx+b… standard form is just a different way of displaying the equation.

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!

# Convert Slope-Intercept to Standard Form Equations

I recently put up a lengthy post that described how to go about performing point-slope form to standard form equation conversions, for when you are trying to express the equation of your line in a specific manner. However, what about if you don’t have an equation in point-slope form, but rather have it in slope-intercept form instead, and now want to write that in standard form? Thankfully, the principles of the algebraic rearrangement remain the same, so if you understand how to do one conversion, the other should be a similar-tasting piece of cake. I’ll outline the steps you can follow in this post so you can convert slope-intercept to standard form. Continue reading Convert Slope-Intercept to Standard Form Equations

# Graphing Parallel Lines

By now, you are probably well on your way to becoming a master of graphing lines! If you understand the information you can get from the equation of a line, graphing is quite easy. However, can you go beyond the given equation? I explained in a previous post about how to graph perpendicular lines, but do you know all about graphing parallel lines as well? Luckily, there is only one key concept to understand, and in this post, I will explain it for you. Continue reading Graphing Parallel Lines

# Point-Slope Form to Standard Form Conversions

In this post, I’m going to explain a very frequently requested topic – how to convert your equation of a line from point-slope form to standard form. Sounds easy, right? Well, it isn’t difficult at all, provided that you understand the terminology and know what you’re doing. Follow along and hopefully all will become clear! Continue reading Point-Slope Form to Standard Form Conversions

# How to Write an Equation of the Line

Following hot on the heels of last time, where I showed you how to find the slope of a line, this time I’m going to build on that thought by describing a process to write an equation of the line. As I explained, just figuring out the slope of your line isn’t enough to specifically describe you line. You actually need some way of anchoring this line to a fixed point on the graph. You could take your sloped line and slide it all over the graph anywhere that you’d like, and you could still wind up with the exact same slope but a completely different formula for the line. That’s because it’s not rooted to any particular place. Luckily, there are very few things required to accurately and completely describe a straight line, and they aren’t that hard to obtain. Continue reading How to Write an Equation of the Line