Use specific values for x and y to show that, in general, 1/x + 1...

Question

Use specific values for x and y to show that, in general, 1/x + 1/y is not equivalent to 1 / x + y.

Accepted Answer

Hello everyone. We're going to solve for the exact values of H. And K. To prove that the expression to over H plus two over K is not equal to the expression to over H plus K. In order to find the exact values. We actually are going to solve this like they are equal. So that way we could see why this wouldn't work. So we're starting out with two over H Plus two over K equals two over H plus K. On my left hand side. I want to create a common denominator so I can combine those fractions. My common denominator is gonna be HK So to make that happen I multiply top and bottom by K. In my first fraction. So I get two K over Hk in my second fraction I need to multiply top and bottom by H. So I get to H over H. K equals two over H plus K. Next I want to take those two fractions on the left hand side and make them into one fraction. So I get two K plus two H over hk equals two over H plus que I noticed that I have the ability factor two out of the first fact fraction. So I'm going to do that. Target two times H plus K. Over hk equals two over H plus K. At this point I notice I have two fractions that kind of look like they're a proportion and I recall when I have something set up like this, I can cross multiply so that's what I'm gonna do. I'm multiplying across the equal sign. So I get to hk and then going in the opposite direction I get two times H plus K. Times H plus K. I want to get the constant away from the variables. So I'm gonna just divide both sides by two and that cancels out. So right now I have each K. Equals and I noticed what I have left on my right hand side are a binomial square. I have H. Plus K. Times H. Plus K. So I recall that that means I can square the first term of H squared square the last term of K squared. Take the two terms multiplied together and times two. So H. Times K. Times two gives me two H. K. So it looks like I'm gonna need to use the quadratic formula to solve this. And in order to use the quadratic formula, my equation needs to equal zero. So I'm gonna subtract each K from both sides. So I'm left with zero equals H squared plus H. K. Plus k squared. So I mentioned the quadratic formula and I'm actually gonna write it out because sometimes people have memorized sometimes not but it is X. Equals negative B plus or minus the square root of B squared minus four A. C. All over two. A. And traditionally when we're using this our equation looks like zero equals a. X squared plus Bx plus C. Well the equation we're using isn't set up quite as traditionally so I want to make a key so I know what I'm plugging in where I'm gonna start out by solving for H. So X. In this case is going to be a church A. Is what's in front of X squared. So what's in front of H squared is one. So A equals one. We're going to treat the letter K as a constant in this case. So B. Is the value in front of X. So in this case the value that's with H. Is K. And C. Is the value that is the last term by itself. And in this case is K squared. We're gonna take this information and plug it into the quadratic formula to find the value of H. So H equals negative B. So negative K plus or minus the square root of B squared. So K squared minus four times A. Which is one times C. Which is K squared all over two. A. So two times one. Alright we've got it all in there now we need to clean it up so the negative K stays put plus or minus under my radical. K squared can stay as it is undue negative four times one is still negative four times K squared. Gives me minus four K squared over to under the radical. I recognize that I have like terms so I'm gonna combine them so H equals negative K plus or minus the square root of negative three K squared over to. Alright next I want to see what I can get rid of out of the radical. Um I recognize that I have a negative under that radical. So I could factor out the square root of -1. As an eye. I'm going to do that so H. Equals negative K. Plus or minus. And I said I have an eye that I can take out. Um The three. There's nothing I can do with, it's gonna stay the square to three. Um Can I take the square root of K. Squared? Yes I can and I will get K. I want to put in front of the eye over to now looking at it. I think that's all I can do. That's gonna be my exact value for H. So we're halfway there so we have negative K. Plus or minus K. I. Radical. 3/2. That is our exact value of H. But we are not done just yet. We need to still find the exact value of K. So I'm gonna scroll down to give myself a little bit of space but I want to still be able to see our equation. So hold on a little too far. There we go. So we're gonna do the quadratic formula again. But this time it's gonna be under this line we're gonna say X. Is K. So looking at it as X's K. We know that A. Has to be the value next to K. Squared which is one B. Is the value next to plain old K. Which is H. And C. Has to be the value that would be the constant in this case it's K. H squared. Alright so we're gonna plug it back into that quadratic formula K equals negative B. So negative H. Plus or minus the square root of B squared. So H squared minus four times A. Which is one times C. Which is H squared Over two times a. Which is one right now. It looks very similar to our previous half of this problem which is a good thing. So K. Equals negative H. Plus or minus the square root of H squared minus let's say four times one is four H squared over to next. I'm cleaning up what's under the radical. It's going over here so K. Equals negative H. Plus or minus the square root of negative three H squared over to. And just like we did with the case in the previous half of this problem We're going to factor out a negative one and be able to pull out an eye because the square root of -1 is I. And we can also take the square root of H squared and get H. three remains under the radical two is my denominator and that's going to be the other half of my answer. So Kay has equal negative H plus or minus the square root. I'm sorry. Plus your minus H. I. Radical 3/2. So we have our two answers scroll back up to our answer choices and this is answer choice. A thanks for coming along for that ride. Have a good day.

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Use specific values for x and y to show that, in general, 1/x + 1/y is not equivalent to 1 / x + y.