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Clarifying standard form rules

Video transcript

- [Instructor] 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. So things that are in standard form would include things like three x plus four y is equal to 10, or two x plus five y 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 three x is equal to negative four y plus 10. Even though these are equivalent equations, this is just not in standard form. Similarly, if I wrote that y is equal to three times x plus seven, this is also not in standard form. Now the place where some people might disagree is if you were to see something like six x plus eight y 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, six, eight and 20, they're all divisible by two, 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, and 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, Ax plus By is equal to C. When you do the exercises on the Khan Academy, it's not going to be checking whether these coefficients, A, B and C are divisible, 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 three x minus four y 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 zero, while here it is less than zero. 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 1.25 x plus 5.50 y 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, and 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, because negative three is our A, and also having coefficients, having our As, Bs and Cs having shared factors, we thing all of that's 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 some real life and we try to construct equations, and based on the information in 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 factor between A, B and C, that A is greater than zero, and that A, B and C need to all be integers, but Khan Academy does not hold you to that.