yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Rational equations intro | Algebra 2 | Khan Academy


2m read
·Nov 10, 2024

  • [Instructor] Let's say we wanna solve the following equation for x. We have x plus one over nine minus x is equal to 2/3. Pause this video and see if you can try this before we work through it together.

All right, now let's work through this together. Now, the first thing that we might wanna do, there's several ways that you could approach this, but the thing I like to do is get rid of this x here in the denominator.

And the easiest way I can think of doing that is by multiplying both sides of this equation by nine minus x. Now, when you do that, it's important that you then put the qualifier that the x cannot be equal to the value that would have made this denominator zero 'cause clearly if somehow you do all this algebraic manipulation and you got x is equal to nine, that still wouldn't be a valid solution. 'Cause if you were to substitute nine back into the original equation, you'd be dividing by zero in the denominator.

So, let's just put that right over here, x cannot be equal to nine. And so then, we can safely move ahead with our algebraic manipulations. So on the left-hand side, as long as x does not equal nine, if we multiply and divide by nine minus x, they cancel out, and we'll just be left with an x plus one.

And on the right-hand side, if you multiply 2/3 times nine minus x, we get 2/3 times nine is six and then 2/3 times negative x is negative 2/3 x. And once again, let's remind ourselves that x cannot be equal to nine.

And then we can get all of our x's on the same side, so let's put that on the left. So let's add 2/3 x to both sides. So plus 2/3, 2/3 x plus 2/3 x, and then, what do we have? Well, on the left-hand side we have one x, which is the same thing as 3/3 x plus 2/3 x is going to give us 5/3 x plus one is equal to six, and then these characters cancel out.

And then we can just subtract one from both sides, and we get 5/3 x, 5/3 x is equal to five. And then, last but not least, we can multiply both sides of this equation times the reciprocal of 5/3 which is of course 3/5. I'm doing that so I just have an x isolated on the left-hand side.

So times 3/5, and we are left with 3/5 times 5/3 is of course equal to one. So we're left with x is equal to five times 3/5 is three. And so we're feeling pretty good about x equals three, but we have to make sure that that's consistent with our original expression.

Well, if we look up here, or if you substitute back x equals three, you don't get a zero in the denominator; x is not equal to nine. X equals three is consistent with that. So we should feel good about our solution. If we did all this algebraic manipulation and we get x is equal to nine, then that still wouldn't be a valid solution because it would have made the original expression on the left be undefined.

More Articles

View All
15 Investments Rich People Make The Poor Know Nothing About
Rich people are making bank in ways everyone else isn’t even aware of. If you watch Until the End, you’ll realistically learn more about money in this video than you did in an MBA course. Here are 15 investments rich people make the poor know nothing abou…
International Space Station Tour on Earth (1g) - Smarter Every Day 141
Hey, it’s me, Destin. Welcome back to Smarter Every Day. I want to be an astronaut, so I love taking tours of the International Space Station online. But there’s a problem. Every time I do this, I can’t get my bearings. It’s like, without gravity, my main…
TIL: How to Transform Mars into Our Second Home | Today I Learned
Hey there, would you like to live on Mars? That’s a garbage idea! If you try to go out there right now, you would simultaneously freeze and choke to death. I’m Brendan Mullin, an emerging explorer with National Geographic and an astrobiologist. I’m here …
A Larvae Lunch | Primal Survivor
This rotting tree becomes a food source for insects, and they in turn might provide a meal for me. There, right there is exactly what I’m looking for. These are just crawling out of it as I’m cutting open this log. There could be hundreds of these inside.…
Ray Dalio on his Principles for Success
Can you think about, like, what are some of your first principles? Life is a journey. It’s an adventure, an adventurous journey in which you come into it with having a certain nature. That nature, um, you’re in a journey to find the path to match up with …
Setting Up Camp in a Tree | The Great Human Race
2.4 million years ago, before humans had weapons or fire, Homo habilis retreated into the safety of trees to escape predators at night. Sounds almost like a hyena. “We have like minutes left really. I think it’s high enough.” “I mean, are you stable tho…