solve-each-system-of-equations-beginalignedxyz6x-yz2x-y-z-4endaligned

Question

Solve each system of equations.
$$\begin{aligned}&x+y+z=6\&x-y+z=2\&x-y-z=-4\end{aligned}$$

Jake Zanazzi · Accepted Answer

Okay, So here&#x27;s their system operations. These three variables three announced. So we know we can solve this using substitution. However, this doesn&#x27;t give us one equation with less than three variables. All equations have three variables. So what we have to do here, this all for one variable in terms of the other two. Plug in that substitution into the second equation salt for one of the variables. And then you have to go back and do it again. Um, that didn&#x27;t make sense. Would go through step by step until makes more sense as we do it. So first, let&#x27;s use this spirit equation and solve for X In terms of y and Z, it&#x27;s gonna give us ax equals why, plus Z minus four. So now we have equation an equation for acts in terms of why and Z we can plug that into this equation and then solve for just why in terms of C So we&#x27;re gonna have acts which who knows why, plus a C minus four minus y plus Z equals two. And if we saw bits for why we&#x27;re going to get I don&#x27;t see, we have Oh, sorry. If these wise they&#x27;re gonna actually cancel. Then we&#x27;re gonna be left with free with that four over. That&#x27;ll kind of six. This is gonna be to Z equals six Z. It&#x27;s just gonna be equal to three. Okay, so now what we can do is plug in the Z into this equation right here. One for acts, right? This is gonna give us why. Plus three minus fourth war lime on this one. Now where you could lose, you have act. You have Z. You have what X is in terms of. Why, So you can plug that into this first equation or really any of them? Let&#x27;s use that first equities Another line minus one. That&#x27;s why. What does he usually knows? Three Equal. Six. This is gonna give us to why, to people six or two equals four. So why is gonna equal to? And now we have why we could get X Another X equals y minus one equals two. Nice one. Because one now we have our orders. Triple external. Why? Policy? People&#x27;s one comma. Two out of three. And now let&#x27;s plug this in just to make sure that this solution actually works, we have X plus y plus equal +61 plus two is three plus +36 x one minus y. That&#x27;s going negative. One plus Z plus three equals two. So that works now if X minus y my next Z one minus two minus three equals negative four. So the solution holds.

Ashley Volpe · Answer

right. So for this equation, I&#x27;m going to take my first equation X Monets. Why, Monta Z equals two. I&#x27;m gonna pair it with my second equation. Negative, X minus Y plus Z equals four. When I add those two equations together, I&#x27;m going to get negative two y equal to six. And so then why equals negative? Three gonna take two additional. I&#x27;m gonna take my 1st 1 but this time I&#x27;m gonna pair it with the 30 equation. I can combine those two equations by adding I will get negative to Z equals eight and so z than equals negative. For Since I know y and Z, I just need to find X And I&#x27;m gonna do that by plugging into the original equation going to use the 1st 1 x minus. Wa Monocacy equals two. And so then when you blowing in, you get X blustery. Since it&#x27;s minus negative and plus four again, minus negative equals two, three, and 47 And when used to track from two, we get negative. So my answer here is negative. Negative three and negative over

Vicki Stebbins · Answer

Freddie. So this should be a fun one. Okay, the and see Nereo, you go by. Oh, let&#x27;s see what I want to get rid of here. Why don&#x27;t we simplify for so we do that? So a we have one x to next. It&#x27;s two x los one. Why t wise Teoh waas one c to seize six. Uh, when that would simplified X plus Why Plus said he cools three. What makes that easier? Now let&#x27;s sit without be so be given X and a negative X and they&#x27;ll zero out even negative. Why? In a wise though zero out and negative z and Z So that&#x27;s heroes out so that he proposed zero equals C room and then see his ex bust lie plus two z equals rooms four. So let&#x27;s put I and C together. What a unique thought since those were the only ones that we can put together, right, So x plus my plus two z equals four and I&#x27;m just going to get all these. Shall we make it easier? Uh, so this Ciro&#x27;s out. This sea rose out and this becomes negative. Z equals negative one, so Z equals one. So now We need to figure out what we have here. I&#x27;m gonna put it into Okay, so two X plus two Why equals offside? Put my two terms issued. Plus two comes. One equals six. So two X was to why plus two equal six. So two x it was too. Why pickles or some X plus y equals two. And so X is gonna equal. I&#x27;m sorry. So why? Because we would have a softer X. So why is gonna equal to minus X? Because I can&#x27;t do anything with that one. Just these be we&#x27;ve already complaint could be combined and see So X And then why is always gonna be two minus X? So you have a wop unload of points where they intersect there

Jake Zanazzi · Answer

right. So this problem were given this system of equations, it&#x27;s these 1st 3 equations right here. Don&#x27;t not this two x or three. Like was he X plus y plus Z equals two X plus two y minus Z equals +62 x plus y minus Z equals five. So my first steps actually gonna be to solve for Z in this equation? I used this first equation are here. I saw frizzy tu minus y minus X. And when I did, was I plug this to minus y minus axe into these 2nd 2 equations right here. So I went to, um Sorry X plus two y minus to plus y plus X equals two x plus three Y uh equals eight when you moved that to over and then you have a two x plus y. My zeke was five. Um, when you played that Z and you end up getting three x plus two, I equals seven. Now we have these two equations with two unknowns, will consult tracks and why we can say that to acts equals B minus three. Why so X is gonna equal four finest three halves? Why get rid of this And if we put that into our second equation with three times acts, which is four minus three house Why plus two I equals seven. If we multiplies out, we&#x27;re gonna 12 minus 9/2 y plus for over two. Why close seven. And once we saw this, we&#x27;re going to have negative five over to fly. It was negative. Five. If you divide negative five, founded by negative five over to get why equals two. But this wife was too back in ah to this equation are here. You&#x27;re going to get acts equals one. So now it&#x27;s pointed these in to our two equations up here to make sure our ex wives make sense. We have two times acts that&#x27;s too plus three times y six equals eight. Then we have three times acts. It&#x27;s gonna three plus two times why that&#x27;s four equal seven. So that makes sense and delete all this work. And we have that life was to annex equals walk. Now we can just use this first equation to solve for Z, we have acts of just two beside one. That&#x27;s why with just two plus Z equals two, this is gonna give us C equals negative one. So now we have our ordered pair acts. Why? Z equals one to negative one person. Preface, ease! And we have our answer.

Rebecca Polasek · Answer

so to solve the system of equation, we aren&#x27;t giving a particular method, but solving would mean we&#x27;re looking for the X, Y and Z that make each of these equations true. So one method that we can do is elimination. And just like elimination was with two equations you&#x27;re looking for a variable to cancel out. So, for instance, if I put together lines one and two equations wanted to I can see that when my variables is going to cancel up because I have a plus Z in a minus Z. I would put these together. Then I would get three x minus y equals one disease than wine minus one on those ease is going to cancel. But because there&#x27;s three equations unlike a system of two equations, I&#x27;m gonna have to do this step twice because I don&#x27;t want to have one equation with two variables and then still another equation with three. So actually, I&#x27;m gonna put together equations one and three as well, because again, I see that that&#x27;s going to cancel disease. So putting together lines one and three, I&#x27;m gonna have four X plus two wide disease Aaron, cancel And then I have eight as that constant number. So now we&#x27;ve created a system of two equations so we can use whatever method we want to solve. Given the equations three x minus y equals one and four x plus two y equals eight. So maybe I want to use substitution. Or maybe I want to use elimination again. But other good options substitution you would get something by itself. Elimination. Obviously looking to cancel one of the variables I do notice if we were to multiply this first equation Times two, we&#x27;d have six X minus two y equals two. And this would work out nicely for one more elimination if I were to put it, then with that other results for X Plus two y equals eight because these wise we&#x27;re gonna cancel up. So we get 10 X equals 10 which would tell me that X equals one. And now that I know what X equals, I can take that into one of my equation options here. Like maybe I take that into the 1st 1 So three times one minus y equals one so minus y equals negative, too, because I would have to subtract that three over and then flipping the sign why would equal to? And now that I have an answer for X and why I can feel this into any of the original equations, I&#x27;m gonna recommend equation one, because it just looks like the simplest. And I&#x27;m gonna fill in the X and the Y that I know and solve regime. So if X is one, why is to Z? I don&#x27;t know Then that gives me three plus Z equal six. So if I subtract that three over to the other side, then I know Z equals three. So that gives me a solution of 1 to 3 writing that as X comma y commas e.

Louisa Benatovich · Answer

So here we&#x27;ve got a system of three infections with three variables as it goes, Why X? And we&#x27;re trying to figure out what each of these is equal to. First step. Choose a variable that we want to cancel. Well, when you look at these three columns of variables, looks like the X is going to be easiest. You can already see the three and the negative three. That&#x27;s how you do math strategizing. So now that we&#x27;ve chosen our victim, variable, you look at these three equations, we choose which ones we want to add together first. Well, like we said, we got the three and the negative three. So how about we try adding these two together first? Because we know we won&#x27;t have to do and multiplication. So let&#x27;s rewrite thumb the X plus for why minus three times three 6 12 c. Perfect. We&#x27;re adding them together. No multiplication necessary. Please cancel as we wanted them to for why minus six wives negative two y. This is a positive 40 is too perfect. We&#x27;ll save this for later. Got a two variable equation going back to the top. We need to pick two other equations that together How about the 1st 1 and the last one? Go down or rewrite? Um, we&#x27;ve got six X plus two y plus four easy. It was to negative three X minus six. Why class 12 0 people&#x27;s five. And what can we do to the bottom equation? To get it to cancel the top equation? Well, we&#x27;ve got six x here, and we have negative three X here. What can we do to get this to negative six x? You guessed it. It are multiplying by just a regular old, too. So when we re write this, get negative six sex perfect battle. Cancel minus 12 Y plus 24 g. Perfect. Now this is gone. Imagine this completely gone, and we&#x27;ll just add this top to the bottom. So put a big plus sign there. All right, six x minus six x, but cancels zero Exactly what we wanted. Do I minus 12. Why is negative? 10. Four Zia plus 24 0 28 z and then to just 10 minutes. Perfect. We&#x27;ll save this for later. So we&#x27;ll go back and look at the top. Are other equation with two variables negative. Two y plus four. Z equals two will write it down here. Negative. Teoh plus four z equals two. So we need to add these together to cancel them somehow. And here it&#x27;s pretty complicated already. So you think this pick either variable? How about we go with the why? What killing? Multiply this by to get it to cancel. Negative 10. Why? Well, we would need it to be a positive 10 right? So what do you multiply? Negative to buy to get a positive 10? Yes, it&#x27;s a negative. Five negative times. A negative is a positive. So we multiply both sides by negative five. Where do you get a positive 10? You get enough in 2010. So this is gone because we have the new and improved version of it. You don&#x27;t usually have to cross the night. I just do that so that you don&#x27;t get confused. I got him a big, big plus sign cause we&#x27;re add even together. So what happens when we adding together? All the Thames can see just that We wanted them to This becomes a Z and then this 12 minus 10 becomes too well. This is interesting to get Z by itself. We divide by eight on both sides. Is he going to over eight? That&#x27;s a fraction. Remember to always simplify the fraction because we can divide both the top and the bottom by two. It becomes one for so easy is 1/4. We have 1/3 of the puzzle finished. So let&#x27;s look back to one of our two variable equations. How about negative 10? Why plus 28 z? People swell because we know what Z is. We can plug it into this equation right here and then sold for a while. All right, so let&#x27;s plug it in. 10. They get 10. Why? Plus 28 times 1/4 equals 12. What&#x27;s 28 times 1/4 It&#x27;s seven. Seven equals 12. You keep going down solving? Sure. Why not get a 10? Why equals five and then you devised by 10 on both sides using your algebra brains? Why equals negative 1/2? Because you&#x27;re simplifying this perfect Z equals 1/4 y equals 1/2. We only have to figure out X. Let&#x27;s go back to the beginning and choose one of our regular old equations. How a bout Negative three x minus six y plus 12 z equals five. It doesn&#x27;t really matter which one you choose. They&#x27;ll all work, so we don&#x27;t know what accessible. Leave that minus six times Why? Which is negative. 1/2 was 12 times Z, which is 1/4 here. There once he is 1/4 wise. 1/2 equals five soldiers. A multiplication negative three x is the same negative six times negative 1/2 is a positive three. 12 times 1/4 is also three equals five. That&#x27;s the same as saying Negative three X plus six. Those add together people&#x27;s fine. Subtract the six over negative one. Divide both sides by negative three. Remember that the negatives air going to cancel So you get X equals 1/3. It causes 1/3. So you did it executes 1/3 z equals 1/4 y equals negative With that nice work

Problem

Solve each system of equations. $$\begin{aligned}…

Problem 11 Medium Difficulty

Solve each system of equations.
$$\begin{aligned}&x+y+z=6\\&x-y+z=2\\&x-y-z=-4\end{aligned}$$

Answer

$\{(1,2,3)\}$

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Mark Dugopolski

Algebra for College Students

Ahyeon H.

Shubham P.

Martha R.

Heather Z.

Solving Absolute Value Inequalities

Rewrite Equations and Formulas - Example 2

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$\{(1,2,3)\}$

Algebra for College Students

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Solving Absolute Value Inequalities

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Mark DugopolskiAlgebra for College Students

Ahyeon H.

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Heather Z.

Mark Dugopolski

Algebra for College Students