Converting from slope-intercept to standard form

- Rewrite Y is equal to 2/3x plus 4/7 in standard form. The equation must be simplified, which means all numbers must be integers that do not share a common factor other than one. Alright, we'll worry about this second part in a little bit. Let's see if we can rewrite this. So it's Y is equal to 2/3x plus 4/7. Let me get my scratch pad out. So it's Y is equal to 2/3x plus 4/7. So the way that it's written right now, this is slope intercept form. It's written in the form Y is equal to mx plus b, where m in this case is 2/3 and b is 4/7. It's very easy to figure out what the slope and what the Y intercept is from this equation. But we wanna write this in standard form. Which would be the form Ax plus By is equal to C. And that extra text they were saying, the equation must be simplified. Which means all numbers must be integers that do not share a common factor other than one. That just means that A, B, and C in the standard form they want need to be integers. And they want them to not have any common factors. So if we got to the point of say 4x plus 2y is equal to 10, well this number, and this number, and this number are all divisible by two. They all have the common factor of two. So we would want to simplify it more. Divide them all by two and then you would get two so you divide this by two. You get 2x, divide this by two you'll get plus y is equal to 5. So this is the form that they're asking for and probably because it's just easier for the site to know that this is the right answer. Because there's obviously a bunch of forms in this way. So let's see if we can do that. Let's see if we can write it in standard form. So the first thing I would wanna do is, well there's a bunch of ways that you could approach it. The first thing you could try to do is well let's get rid of all of these fractions. And the best way to get rid of the fractions is to multiple by three and to multiply by seven. If you multiply by three you get rid of this fraction. If you multiply by seven you get rid of this fraction. So if you multiply three and you multiply by seven. Let me just rewrite it over here. There's actually also a couple of ways that we can do this. So if you multiply. So one way to do it. So we start with Y is equal to 2/3x plus 4/7. So if I multiply this side by three and I multiply by seven, I have to do that to this side as well. So this is going to be multiplied by three and multiplied by seven. So the left hand side becomes 21y. 21y. Three times seven of course is 21, we just figured that out. We would distribute the 21. 21 times 2/3, well let's see. 21 divided by three is seven, times two is 14. So it's gonna be 14x. And then 21 divided by seven is three times four is 12. So just like that I was able to get rid of the fractions. And now I wanna get all the X's and Y's on one side. So I wanna get this 14x onto the left side. So let's see if I can do that. So I'm gonna do that by. To get rid of this I would want to subtract 14x. I can't just do it on the right hand side I have to do it on the left hand side as well. So I wanna subtract 14x, and then what am I left with? Let me give myself a little bit more space. So on the left hand side I have negative 14x plus 21y. Plus 21y is equal to. Let's see and I subtracted 14x to get rid of this. And then I have this is equal to 12. Now let's see, am I done? Do these share any, do 14, 21, and 12 share any common factors? Let's see, 14 is divisible by two and seven. 21 is divisible by three and seven. 12 is divisible by two, six, three, four. But all of these aren't divisible by the same number. 14, let's see. 14 is divisible two, so is 12, but 21 isn't. 14 is divisible by seven, so is 21, but 12 isn't. And 21 and 12 are divisible by three but 14 isn't. So I think this is about as simplified as I could get. If there was a common factor for all three of these numbers then I would divide all of them the way I did in that previous example. But that's not the case right over here. So it's negative 14x plus 21y is equal to 12. So let me see if I can remember that and type that in. So it is, negative 14x plus 21y is equal to 12. Now let's see if I got it right. It worked out.