Clarifying standard form rules

We've talked about the idea of standard form of a linear equation in other videos, and the point of this video is to clarify something and resolve some differences that you might see in different classes in terms of what standard form is.

So everyone agrees that standard form is generally a linear equation where you have some number times x plus some number times y is equal to some number. Things that are in standard form would include things like 3x plus 4y is equal to 10 or 2x plus 5y is equal to negative 10.

Everyone would agree that these are standard form, and everyone would agree that the following are not standard form. So if I were to write 3x is equal to negative 4y plus 10, even though these are equivalent equations, this is just not in standard form. Similarly, if I wrote that y is equal to 3 times x plus 7, this is also not in standard form.

Now, the place where some people might disagree is if you were to see something like 6x plus 8y is equal to 20. Now, why would some folks argue that this is not standard form? Well, for some folks, they would say standard form the coefficients on x and y and our constant term, so our a, b, and c can't share any common factors.

Here, 6, 8, and 20, they are all divisible by 2. And so, some folks would argue that this is not standard form, and to get it into standard form, you would divide all of these by two. If you did, you would get this equation here.

Now, that's useful because then you only have one unique equation, but on Khan Academy, we do not restrict in that way, and that is also a very popular way of thinking about it. We just want you to think about it in this form: a x plus b y is equal to c. When you do the exercises on Khan Academy, it's not going to be checking whether these coefficients a, b, and c are divisible and have a common factor.

So for Khan Academy purposes, this is considered standard form. Although, don't be surprised if you encounter some folks who say, "No, we would rather you remove any common factors."

Now, another example would be something like negative 3x minus 4y is equal to negative 10. So some folks would argue that this is not standard form because they want to see this first coefficient right over here, the a, being greater than 0. While here, it is less than 0.

For our purposes on Khan Academy, we do consider this standard form, but I'm just letting you know because some folks might not because this leading coefficient is not greater than zero.

Now, another example that some people might be on the edge with would be something like 1.25x plus 5.50y is equal to 10.5. The reason why some people might not consider this standard form is that a, b, and c are not integers. Some folks would say to be in standard form, a, b, and c need to be integers.

You could multiply both sides of the equation by some value that will give you integers for a, b, and c, but for Khan Academy purposes, we do consider this to be in standard form. We think this is important actually, not just being able to have non-integers as a, b, or c, but also being able to have a negative a right over there; this negative 3 is our a.

Also, having coefficients having our a's, b's, and c's shared factors we think all of that is important because sometimes the equation itself has meaning when you write it that way. We'll see that when we do some word problems. When we actually go into real life and we try to construct equations based on the information, the equation, it's easier to understand if you keep it in this form.

So for Khan Academy purposes, this is all standard form, but it's good to be aware in your mathematical lives that some folks might want to see the restriction of no common factors between a, b, and c, that a is greater than 0, and that a, b, and c need to all be integers. But Khan Academy does not hold you to that.