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Equations with one rational expression


Video transcript

So we have 14x plus 4 over negative 3x minus 2 is equal to 8. And I'll give you a few moments to see if you can tackle it on your own. So this equation right here, at first it doesn't look like a straightforward linear equation. We have one expression on top of another expression. But as we'll see, we can simplify this to turn it into a linear equation. So the first thing that I want to do is, I don't like this negative 3x plus 2 sitting here in the denominator, it makes me stressed. So I want to multiply both sides of this equation times negative 3x minus 2. What does that do for us? Well, on the left-hand side, you have this negative 3x minus 2, it's going to be over negative 3x minus 2, they will cancel out. And so you're left with, on the left-hand side, your 14x plus 4. And on your right-hand side, you just have to multiply 8 times negative 3x minus 2. So you are left with-- well, 8 times negative 3x is negative 24x, and then 8 times negative 2 is negative 16. And there you have it. We have simplified this to just a traditional linear equation. We've got variables on both sides, so we can just keep simplifying. So the first thing I want to do, let's just say we want to put all of our x terms on the left-hand side. So I want to get rid of this negative 24x right over here. So the best way to do that, I'm going to add 24x to the right-hand side. I can't just do it to the right-hand side, I have to also do it to the left-hand side. And so I am left with, on the left-hand side, 14x plus 24x is 38x. And then I have the plus 4-- Is equal to, well, negative 24x plus 24x. Those cancel each other out, and we are left with just the negative 16. Now we just have to get rid of this 4 here. Let's subtract that 4 from both sides. And we are left with-- and this is the home stretch now-- we are left with 38x is equal to negative 16 minus 4 is negative 20. And so we can divide both sides of this equation by 38. And we are left with x is equal to negative 20 over 38, which can be simplified further. Both the numerator the denominator is divisible by 2. So let's divide the numerator and denominator by 2, and we get negative 10 over 19. x is equal to negative 10 over 19, and we are done. I encourage you to validate this for yourself. It's a little bit of a hairy number right over here, but to take this number, substitute it back into our original equation, and validate that it actually does satisfy that equation.