in-3-20-solve-each-equation-and-check-fraca-14frac8a3

Question

In $3-20,$ solve each equation and check.
$$
\frac{a-1}{4}=\frac{8}{a+3}
$$

Dilip Paruchuri · Accepted Answer

All right. Hey, guys. So, in this problem, we have a minus one over four equals eight over eight plus three. Our goal is to solve for a and verify that are solutions air correct. So let&#x27;s start. We&#x27;re gonna start by getting our our eight terms on all out of the denominator. So we&#x27;re gonna do is we&#x27;re gonna multiply by a plus three. The quantity of April&#x27;s to be on both sides on our goal is to get the right hand side without an A A term in the denominator. And that&#x27;s what we&#x27;re gonna achieve the April stories. Cancel On the right hand side, We&#x27;re just left with the value eight on our left hand side. We have a plus three times the quantity a minus one over four. Just so we make this problem a little bit easier for ourselves. What I&#x27;m gonna do, I&#x27;m gonna cancel out the four from the denominator on the left side as well. Basically wants by four on both sides getting rid of the dot the denominator on the left hand side as well. So the forest cancel out. We&#x27;re left with a plus. Three times a minus one equals 32. And let&#x27;s simplify this by simplifying mean expand the left hand side. We&#x27;re gonna get a square minus a plus three a minus three that&#x27;s going to equal 32. Let&#x27;s simplify this part. We have a square plus two. A minus three equals 32 and subtract 32 from both sides. We get a squared plus two a minus 35 minus 35 this up and minus 35 is equal to zero. So let&#x27;s bring this up all the way here, so it&#x27;s easier to solve. We have more room a square close to a minus 35. Factoring that we&#x27;re going to get a plus seven times a minus five, that&#x27;s gonna equal zero. So we know that a plus seven equals zero or a minus five is equal to zero. So our solution said, is gonna be a equals five comma minus seven. And let&#x27;s verify this on the second page. So the equation that we&#x27;re trying to verify is a minus. One over four equals eight over a plus three value. The first value. Let&#x27;s try to verify a equals 25 because our two values are five and five and negative seven. So our first value will verify is deposited five. So left hand side. If we plug in a we&#x27;re gonna have five minus one over four five minus one is for for over four is equal to one. And on our right hand side, we&#x27;re gonna have eight over five plus 35 stories. Eights of the denominator is now eight and eight. Over eight is equal to one. Left hand side, right hand side or equal. So we know that&#x27;s true. Equals five verified to be correct. Now let&#x27;s try a equals minus seven. Now on the left hand side, if we plug in a equals minus seven would have minus seven minus one, divided by four. That&#x27;s going to be equal to minus eight over four, which is going to be equal to mine. This too. Then we have our right hand side. We have each over minus seven plus three minus seven plus three is equal to minus 48 Divided by minus four is equal to minus two. We have our left hand side in the right hands out of the same. So a equals minus seven is also a verified solution. So we can definitively say that a was five and a equals minus seven are our correct values. And so it&#x27;s just raised the stray marks right here. So this, uh, thing these solutions in the red box are the answers to the equation. A minus one over four equals eight over April&#x27;s three. Thanks for listening.

Derrick Hanson · Answer

we&#x27;re being asked to solve this equation out for A I think the most important thing to recognize at the beginning is that if you were taking 1/4 times a or 1/2 times A that A is on Lee being multiplied to the numerator, right? If I asked you to represent a as a fraction you&#x27;d represented as a over one. So if we&#x27;re taking 1/4 times essentially a over one, it&#x27;s one times a on top for in the numerator four times, one on bottom in the denominator. So I think really a better way to right this equation would be a over four plus eight equals a over to. I think that&#x27;s the better way to represent this equation to just save for many possible mistakes. Now that we&#x27;ve got the equation shown this way, obviously, the next biggest issue is that we have some fractions here. We have rational expressions I don&#x27;t want to have to solve with fractions. It&#x27;s not fun. I don&#x27;t think many people prefer fractions over whole numbers. It opens us up to more mistakes, and it&#x27;s just gonna make it tough to solve in general. So what, we&#x27;re gonna do you? We&#x27;ve already done a section talking about adding and subtracting rational expressions. Right? And we know we need to get a common denominator. Well, we do want to find a common denominator here as well. But what we&#x27;re going to do is, once we find that common denominator, we&#x27;re going to multiply the entire equation by that denominator in order to get rid of to cancel out all of our denominators. Meaning? I see that I have a four and the two for my denominator. So least common denominator means we&#x27;re looking for the least the lowest number, that both two and four can multiply too. In this case, that would be four right. Four times one is for two times two is four. Now, we&#x27;re not going to separate these fractions and multiply the numerator and denominator by something to try and get them all the same. Instead, we&#x27;re gonna take this entire equation we said our common denominators for So let&#x27;s multiply this equation by four. Meeting 4/1 if you prefer to write it that way. So the point of that is, as long as we multiply every single term by that four, we&#x27;re not changing the value of the equation. And assuming that we found a common denominator correctly, this should cause all of our denominators to go away, to simplify out. Let&#x27;s see, I take four times a over 44 times, ese for a four times. One is four. So we&#x27;d have foray over four take eight times for eight times for is 32. Take a over two times for four times a xfiles or a two times one is two. Okay, so this is the equation we have now that we&#x27;ve multiplied every piece by four. Since we&#x27;ve multiplied every single term by four, it&#x27;s still equivalent. What we started with Look at our two fractions are too rational. Expressions four A over four could certainly simplify to just a 32 is 32. Nothing with that for a over to four, divided by two is to. So that would leave us with two a. So see, what happened here is now. We no longer have any denominators. We don&#x27;t have, you know, rational expressions or fractions. We have a pretty straightforward linear algebra problem. All we got to do is get a by itself, I would say The easiest way to do that would be to subtract this a over to the right side to get it away from 32. So now we just have 32 on the left side. Two a minus a or minus one. A. If you want to think of it that way to a minus one a is simply one A meaning. Our answer would be that a is equal to 32. It&#x27;s all about getting those denominators to cancel out. The solving part after that is usually pretty straightforward.

Derrick Hanson · Answer

trying to solve this equation out for be. I see that I do have one rational expression here. So my first goal is to figure out what a common denominator would be that I could most by this equation by to get to where I have no rational expressions, no fractions. It&#x27;s especially important on this one because the be that we&#x27;re solving for actually happens to be in the denominator. And until we get be out of the denominator, it&#x27;s not gonna be useful to us. Okay, also, well, it&#x27;s worth making a note here that once we do get an answer, we will have to plug that answer back into the original equation that we have sitting here to make sure it really works. Because since we have be in the denominator, it&#x27;s possible that the answer we get could actually cause this whole equation to become undefined, Which means even if we solve out for be correctly, it could still end up being a no solution. So we&#x27;ll have to check at the end to make sure it works. Okay, well, I only have one Fraction, one rational expression and the denominators be not very hard here to find a common denominator then, because there&#x27;s only one denominator. So that&#x27;s what we&#x27;re gonna multiply, meaning we&#x27;re going to multiply by B or be over one if you prefer to think of it that way. Same thing. Either way, we have to multiply every single term in this equation by B so that we aren&#x27;t changing the actual value of the equation. So I take be times 18 that is simply 18 B. Then we take be times four over B and that would be four b Overbey. Then we take be times 10. That would be 10 b. So we can see here then that the one rational expression the one fraction we have does simplify out because be Overbey would cancel. Right? So it be Overbey cancels. That means what we really have is 18 B minus four equals 10 B. That&#x27;s what we&#x27;re actually looking at here. All right, so now that I&#x27;ve gotten rid of all the fractions, the rational expressions, my goal is simply to isolate B and see what it equals. I would say easiest way to do that to start would be subtracting 18 b to the other side of the equation because that combines my beast together while also getting my constant by itself. So I would just haven&#x27;t negative four on the left side. 10 B minus 18 b would be a negative eight b. That&#x27;s great, but I&#x27;m looking for just be so I need to move that negative eight over the opposite of multiplication. His division so we would be dividing by negative eight. B is equal to negative four over negative eight. A negative divided by a negative is always a positive. So that part&#x27;s covered and 4/8 simplifies to 1/2. So it looks like B is equal to 1/2 now just to verify that, going to rewrite the original problem over here, which was 18 minus four Overbey, equals 10. We need to plug 1/2 in for B to make sure that this actually works. So that means we would have 18 minus four over 1/2. A complex fraction equals 10 last section. You would have learned complex fractions, so you know that four divided by 1/2 is actually the same thing as four times to over one or effectively four times two alright, four divided by the fraction 1/2 dividing by a fraction of the same was multiplying by its reciprocal. So you simply flip 1/2 to 2/1, and you can multiply well, four times to would be eight. Meaning you&#x27;d have 18. Minus eight equals 10 18 minus eight is 10. So we get 10 equals 10 checks out. Looks like 1/2 truly is our answer.

Derrick Hanson · Answer

So we&#x27;re trying to solve this problem for X first thing notice here is that obviously we do have some rational expressions here. Some fractions. I don&#x27;t want to leave those those fractions. So the first thing I want to do is I want to find a common denominator so that I can multiply the entire problem by that number to get rid of all of my denominators. Right now, I see a denominator of three and denominator of four. So the question is, what is the least what is the lowest number that both three and four can multiply up to? In this case, I would say that is 12. Okay, so we&#x27;re going to multiply this entire problem then by 12. Understand? This is 12/1, meaning 12 is being multiplied to the numerator. Okay, I have to multiply every single part of the equation by 12 so that I&#x27;m not actually changing the value of this equation. Meaning I&#x27;m going to take 12 times two x over three. Well, 12 times two x would be 24 x over three. Then I have to take 12 times one pretty straightforward. 12 times one is just 12 right, And then finally, I have to take 12 times three x over 4 12 times three x would be 36 x over four. So now, assuming that we chose something that a number that truly was a common denominator, all of my fraction should be able to reduce well. 24 x over three could reduce down to eight. X 12 Doesn&#x27;t need to reduce its on a fraction anyways. And 36 x over four should reduce to nine x 36 divided by four is nine so good? Looks like we did do the right thing here We did pick a common denominator cause all my fractions are gone. So now I simply just need to solve this equation out to get X by itself and figure out what happens equal. Ah, I would say they use this way to do that would be dis attract eight x to the right side because that goes ahead and isolates the Constance as well as the exes at the same time. So that would leave me with just 12 on the left side of the equation. Nine X minus eight X would be one x or just x so looks like our answers. X equals 12. That&#x27;s it.

Dilip Paruchuri · Answer

Hey, guys. So in this equation, we have three over a plus. 1/2 is equal to 11 or five. A. Our first step is to bring all of our is gonna be to bring all of our A terms to one side of the equation. So let&#x27;s start with that by subtracting three over a from both sides. And that&#x27;s gonna give us on our left hand side just 1/2 because of three over Ace, cancel out and our right hand side we&#x27;re gonna have 11 over five a minus three a. And now our next step is gonna be to combine. Oh, that&#x27;s an A, by the way in on eight and A our next step is gonna be combining these 28 terms in tow. One a term. So I want to do that by finding a common denominator or common denominator here is clearly five, eh? So what we&#x27;re gonna do is we&#x27;re gonna do 1/2 is equal to 11 over five day. Don&#x27;t. The denominator is already five day, so we don&#x27;t need to do anything to this one. But for three over a, we&#x27;re gonna multiply by five on the numerator and the denominator and get 15 over five A if you know where. Five. A And so we have the equation supervised toe 1/2 and we know that 11 11 05 a minus 50 or five a is minus for over five day. And now what we&#x27;re going to do is we&#x27;re going to multiply by five a on both sides because we want our A term on the numerator, not the denominator. What&#x27;s the best way to do that? To get it out of the denominator by multiplying by. It&#x27;s reciprocal. So we cancel out the five day and the denominator here in the five that were multiplying by. And let&#x27;s finish this form of here. And so on the left hand side will lift with fi A over to is equal to minus four. And now, if we want to buy buy 2/5 on both sides, we get that our answer to this is gonna be a is equal soon. Negative. Eight. It Yeah, And so now let&#x27;s take a look and let&#x27;s verify it for both The bar, for both of the spokes sides of our equation left hand side first. So we had three over a plus 1/2 three over. A is gonna be such A is going to substitute it in for 1/5. So we&#x27;ll have three divided by 8/5 plus 1/2 and three divided by negative. 8 50 is the same as three times negative. Five over eight Just how you divide fractions and that&#x27;s gonna be plus 1/2 native. 33 times. Notify Will Rate is the same. His negative 15 over eight and our 1/2. Since we know that, we&#x27;ll have to add these to fracture together. 1/2 is the same thing as for eight. We just won&#x27;t buy Buy four on both sides of the equation. That gives us a negative 11 over eight on our right on our left hand side. Sorry, so they could have 11 over eight is water left hand side equals tomb. And now let&#x27;s take a look at our right hand side or right hand. Side is 11 over five, a 11 over five A&#x27;s. Let&#x27;s plug in negative 8/5. So that&#x27;s gonna be 11 over five times. Negative. 1/5. The fives cancel out, so we&#x27;re left with 11 over negative. 11 over eight his right hand side left inside are equal. So we know that a equals negative 8/5 is our answer and it is correct. Thanks for listening. And I hope this hoped with solving this ration.

Derrick Hanson · Answer

in order to solve this problem out for. Why, while we&#x27;re gonna need to do is we&#x27;re gonna need to multiply the entire equation by whatever a common denominator would be for this problem. In order to cancel out all of our denominators. Looking at what I&#x27;ve got here, I have denominator of y plus two and I have a denominator of why times y plus two. While for our common denominator, it needs to be something that every problem can multiply up to meaning. I don&#x27;t want to be missing anything. So I need to be multiplying by both y and y plus two because both dominators have a y plus two in common. But since one of them as a why I have to account for the most amount of stuff. So we need to multiply by a white times y plus two. Okay, if we&#x27;re gonna do that, we have to multiply every single term by that. So if I take why times y plus two times my first fraction. A four over Why? Plus two? Well, the why plus twos would cancel out, leaving me with a four. And why, If I take one times why why plus two? Well, one times anything is just that number. So I get both the why and the why plus two. Not much going there for my last fraction. If I take why Times y plus two and multiply that times eight over why towns lifeless too Well, there&#x27;s a y y plus two on top, and there&#x27;s a y Y plus two on the bottom so they would cancel, leaving us with simply eight. I thought I changed. Now that we have cancelled out all of our denominators, we can actually go about trying to solve this problem out. I do see that I have a distributive property to do here. So we rewrite the four y and for why times, Why would be why squared why times To would be to Why? And we have the minus eight. So now I&#x27;m noticing that I have a y squared, which means we have a quadratic equation that we&#x27;re trying to solve here. So we&#x27;re no longer going to try to get all the wise on one side and all the constants on the other. Now that it&#x27;s a quadratic, we want to get everything on the same side because if we get the equation equal to zero, then we can either try and factor this or we can try and use the quadratic formula, Whatever your personal preference would be. Meaning? I&#x27;m actually going to subtract this for why, To the other side, giving us zero equals Why squared two Y minus four. Why would be negative to why and minus eight. Now you have your choice of whether you would prefer to factor this or whether you would prefer to use the quadratic formula. It truly is up to you. I guess you could also complete the square if you wanted to, but I think most people prefer to either factories the quadratic formula cause those usually seem to take less time than completing the square, but you can choose what you want to do with that. Either way, you should end up with two answers off of this, and then we&#x27;ll have to plug them back in to see which ones work. If you go to factor this two factor this you would be looking for two numbers that multiply too negative. Eight, but add to negative two. We talked about this in the sections previous to this, um, two numbers that multiply Negative eight, but add to negative two would be negative four. And positive, too, since the coefficient in front of the Y squared is just a one I don&#x27;t have any more work to do. We simply have zero equals. Why, minus four as a factor and why, plus two as a factor. This has been fully factored. Now I can move on to the final step to solve of actually solving out for the number zero product, Property says. Since these two factors air being multiplied to each other and the equation is said equals zero, I am allowed to take each of these factors and set them equal to zero separately. So for the first equation, I&#x27;d add for to the other side and you get why equals positive four. For the second equation, I would subtract two to the other side and get why equals negative, too. So right now it looks like your answers are Y equals four and y equals negative, too. But before we say for certain that those are both our final answers, we need to take a look at the original problem, which was four over why plus two equals one minus eight over wide times y plus two. We need to plug both four and negative to infer wine at the same time. Some different times separate several times. If either of them plugs in and causes us to get zero in one of our denominators, that would cause the entire fraction to be undefined, which means that number is not actually a solution. We didn&#x27;t do anything wrong, Algebraic Lee. We still solved out for the correct numbers. It&#x27;s just they may not actually work when we actually plug back in. So let&#x27;s start with 45 Plug in four. I&#x27;d have four over four plus two equals one minus eight over four times the quantity of four plus two. Well, four plus two is six that checks out fine. Four plus two over here is also six and four times six would give me 24 on this one, neither of which is zero. This problem checks out. Four is definitely one of our answers. Now let&#x27;s try the exact same thing for if we plug in negative, too. That means we&#x27;d have four over negative Two plus two equals one minus eight over negative, too. Times negative, too. Plus two. Well, you may have already noticed it, but I can see for my very first fraction negative two plus two is zero four. Divided by zero is undefined. Which means it doesn&#x27;t even matter what I have for the rest. This problem once one part of it&#x27;s undefined. The whole thing is undefined. Which means while we did solve out for negative to correctly when he plugged back in, we realised that negative to does not actually work. Meaning are Onley Answer for this question is positive for

Problem

In $3-20,$ solve each equation and check. $$ \f…

Problem 18 Hard Difficulty

In $3-20,$ solve each equation and check.
$$
\frac{a-1}{4}=\frac{8}{a+3}
$$

Answer

$\{-7,5\}$

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$\{-7,5\}$

Algebra 2 and Trigonometry

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Grace H.

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Ann Xavier Gantert

Algebra 2 and Trigonometry