When you have to determine the equation of a line from a graph, there are a couple of ways that you can express your answer. All of these solutions, essentially, are exactly the same. That is, if you work backwards, you can take each form of the equation and draw the exact same line. The only thing that is different is the way that it looks, but the underlying mathematics are the same. As I mentioned in my last post, one of the more common ways of expressing an equation of a line is in slope-intercept form. In this post, I am going to show you how to write an equation in standard form.
By now, you are probably familiar with this easy-to-remember equation:
This is the slope-intercept equation, where x and y form a coordinate on the line, m represents the slope of the line (think: rise over run), and b is the y-intercept. However, you may instead be asked to express this in standard form, or write a standard form equation. In simple terms, this basically means to write your equation with the x and y variables (and their coefficients) grouped together on one side of the equals sign, with the constant term on the other side.
Perhaps seeing the general version of this equation will be more helpful than words. If we say that x and y are the variables of the graph, and A, B, and C are coefficients, we can assemble these in the following way to show a generic equation in standard form:
You can hopefully notice that this looks very similar to the slope-intercept equation of the line, but things seem to be slightly rearranged. That is the key point that I want you to be able to take away from this post – all of these versions are very similar and are built from the same numbers, but they are just written in a different arrangement from what you may be able to immediately recognize. So, really, being able to write an equation in standard forms comes down to your ability to properly reorganize the terms. Let’s take a look at an example so that you can see what kind of work is required.
Write an Equation in Standard Form
Let’s assume that you are actually given an equation, and you simply need to reorganize it from one form to the other:
As you can see, in this basic example, it was just a matter of grouping the x and y terms on the same side, while leaving the constant value on the other side. Note that I wrote the x value first, followed by the y term, as required by the standard form. You will also often be asked to specifically identify what the A, B, and C values are, so in this case, A is -5, B is 1 (assumed, since there is no explicitly given coefficient), and C is 2. Alternately, you would also be correct if you did the rearrangement the other way – that is, subtracted y from both sides, and then subtracted 2 from both sides. That would give you:
In this case, A is 5, B is -1, and C is -2. Either of these two solutions are correct. Technically, I prefer to have an answer where the x value has a positive coefficient, so this second version is probably the one that I would use.
What about a slightly more complicated-looking starting equation?
In this case, the standard form equation of a line has the following values for its coefficients: A is 1/2, B is 1, C is -7. This is a correct answer, and you could do as I outlined above to move terms the other way and result in flipping all of the signs. However, another step that you could be required to do is to get rid of all numbers in denominators. In this case, the x has the coefficient of 1/2. To get rid of this, you simply have to multiply everything by that value, 2. In that case, you can see that this revised equation is now:
Once again, this form of the equation produces the exact same line as the original form given in the question, despite it appearing to be different. The math works out – graph it out to prove to yourself!
That brings me to another point that I would like to make on this subject. So far, I have explained to you how to write an equation in standard form. However, that assumes that you have the equation written in slope-intercept form, or the actual line itself, from which you can easily pull out values to start with slope-intercept (or another similar) form. What about if you have done the work, converted to standard form, and now want to check that you have the same line as you started with? Or if you are starting with this form of the equation and want to produce a line to show it? Graphing standard form lines is probably the easiest to do if you convert it to something like slope intercept form, and then determine your slope and intercept and easily plot from that data.
Alternately, you could always just produce a table of values, and plot a series of points to help you plot your line. To plot a table of values, all you have to do is substitute in any value for x that you want, and then solve it for the y variable. Do this to obtain several coordinates on the line to gain more confidence in the position of your line.
So, now you know how to write an equation in standard form and in point-intercept form. Hopefully I have been able to convey to you that these different ways of representing the equation of the line still refers to the exact same line. This is easy to prove by plotting a graph from a table of values that you can easily create for both forms of the equations. Using the slope-intercept form of the equation is probably easier and more common simply because of the data that it immediately provides to you – that is, it is immediately obvious what the slope and y-intercept of the line is. Conversely, graphing standard form equations takes a little bit of manipulation of the equation to be able to identify these important graph points. The important point is to realize that, even though one form may need more work, these describe the exact same line.
A final point that I will note here on this post is perhaps something that isn’t immediately obvious. All of the examples I have talked about so far have an x and a y in them, with no exponents. Technically though, they each have an exponent of 1. When you have an equation that only has first degree variables, you are always dealing with a straight line. Second degree variables, or variables with an exponent of 2, will always be parabolas, and higher degree variables will always have particular shape depending on the exponent number. This isn’t something that needs to be considered at this point, where you are just looking at how to write an equation in standard form. But in the future, it will be helpful to immediately recognize what kind of function you have.
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