discussion-make-up-a-system-of-three-linear-equations-in-three-variables-for-which-the-solution-set-

Question

Discussion
Make up a system of three linear equations in three variables for which the solution set is $\{(0,0,0)\} .$ A system with this solution set is called a homogeneous system. Why do you think it is given that name?

Jake Zanazzi · Accepted Answer

right. So this problem we asked to make a system of three linear equations and three variables sports. The solution said 000 To do this, we can just notice that if we set the right side of these equations equals zero, then we&#x27;re gonna have If I was, like ax plus, uh, two y plus three z here, and we have negative X plus five y minus 70 here. When you have, you know, negative four acts plus two y minus Z equals here. Here, +000 is the solution. Uh, we don&#x27;t worry about eliminating or substituting any system of three variables with three equations where the right side include zero. You&#x27;re gonna be able to plug in zero for X Y and Z gets here. So this is called that tells a homogeneous system. Why do you think it&#x27;s given this name? Because you can multiply these numbers by any value. They&#x27;re still gonna have 000 as a solution

Michael Jacobsen · Answer

in this video, we&#x27;re considering what would happen if we have a system of em equations in n unknowns. Let&#x27;s suppose here that n is greater than M know the words. There&#x27;s going to be more variables than there are equations in this system. In this case, the system is under determined in the sense that we have less equations than actual variables in our system. Now let&#x27;s do the following. Let&#x27;s assume this system is consistent. Well, for a consistent system, there is one of two possibilities that could happen. Either will have a unique solution or will have infinitely may solutions. We wonder if the system is under determined and it&#x27;s consistent. Forgot a t here. How many solutions will have? Well, one way to tackle this problem is let&#x27;s let hey be the coefficient matrix for this system. Let&#x27;s sketch roughly what the generic form of the Matrix A would look like. Keeping in mind that the number of equations is greater than the number of Excuse me, The number of equations is fewer than the number of unknowns. That would mean that if we have here M equations, there&#x27;s M Rose. Then if we have an variables. Then there will be altogether and columns. We&#x27;re recall that and is greater than him. So we&#x27;re told that were consistent here. And let&#x27;s look at the best case possibility where we have the most possible pivots. We would have a one here, Ben zeros about. Below that we go 011 then just zeros below that next would be 001 and then zeros below and so on. But because in is bigger than them, eventually what&#x27;s going to happen here is we&#x27;ll get at most a pivot in the very final row and zeros all above. But then that means because and is greater than M, there will be Rose that contain nothing but zeros. This implies that there must be a free variable. The punchline to this argument here is if there is more columns than Rose, then we cannot put a pivot in every single column. But now we know two things. The system is consistent. There&#x27;s a free, very bull, and this gives us an immediate conclusion. There must be in fine nightly, many solutions

Ankit Gupta · Answer

our system off linear equations in three variables. ISS variables is glutinous he x plus b. Why plus see there is because the so therefore it has to be there. 10. In the similar form, it has to video 10 Indo similar for

Nyah Kshatriya · Answer

this problem is asking to describe a system of linear equations in three variables. So you might know that a system in two variables includes and a coefficient times an unknown plus another coefficient times another unknown is equal to some constant. And so in this case we have two unknowns, and this is two variables. So if we have a system of linear equations and three variables, it&#x27;s kind of the same thing. But the general form is a little different. The general form of the equation is going to be a X plus b. Why, plus C Z is equal to D. Where are unknowns? Our X, y and Z? So we have three variables, and then a be C and D are all real numbers. So some sort of constance or coefficient values and so an example of a system of linear equations and three variables is X plus two y minus three Z is equal to nine, and so in this case we have are a value which is just one there&#x27;s. But the coefficient for X is one. We have our be value, which is to R C. Value is negative three and then our devalue is nine and we have are unknowns X, y and Z. So that is a system of linear

Arden Mccarthy · Answer

for this question. We want to find the values of a for which this system equations has no solution. Exactly one solution and infinitely many solutions, So we&#x27;ll start with no solution. So first things first. We want to write out our Matrix from the given equation, so we have one. Next one plus two x two plus x three is equal to two to x one minus two x two plus three x three is equal to one and we have one x one plus two x two minus a squared minus three x three is equal to a So before we go have been started. Let&#x27;s remind ourselves of what the Matrix looks like. That has no solution. So the requirement is that the reduce special inform has a one in the right most column. So let&#x27;s do so matrix simplification. Here, our first step is gonna be extinct, are three and said that equal to our three minus are one. So if we do that first column stays the same. Second column. Cesaire Sorry, First road stays the same. Second row stays the same, and the third row become 00 And then we have negative a squared minus three minus one. And then over here we have a minus two. So what we want to dio is we want to figure out a value such that this is equal to zero, that this is not able to see her because that will give us and inconsistent system of equation. So let&#x27;s first simplify this turn over here. So we have negative a square minus three minus one equal to negative. A squared plus three on this one is equal to a negative. A squared plus two right. And we want it to be equal to zero. So let&#x27;s go ahead and do that now and solve for the A that will make this condition true. So a squared is gonna be able to to because we moved the a square to the other side. Then we take the square root but both sides and we get that a has to equal the square root of two. OK, so that&#x27;s our first requirement. Now let&#x27;s go ahead to our second requirement, which states we want a minus two to not be equal zero. So if a minus two can&#x27;t be able to zero, we want any thio not equal to two. These two requirements agree with each other, since two is not equal to the square to. So we know that for no solution to exist, we want a to be equal to the square of two. But for the next one, for exactly one solution, we know that system equations will have exactly one solution if the reduced excellent form of The Matrix has no free variables. So we&#x27;ll go ahead and we&#x27;ll take this simplified matrix here and we&#x27;ll just go ahead and put it on the other page so we don&#x27;t have to repeat our row operations. So we have 1212 two Negative, too. 31 00 negative. A squared plus two and then a minus two. Okay, so let&#x27;s continue so reduction, so we can get a closer to produce a short form. So to do that, uh, let&#x27;s first take our second row and said it equal to the first row plus the second row. And this will help us knock out this term right here. So when we do that, we get this matrix. First row stays the same second row since we&#x27;re out of it, becomes 30 for three and then the bottom row stays the same. Okay, so now let&#x27;s go ahead and do another row operation. So this row operation will help us knock out this turn right here to make it a zero. So to do that, we want to take our one and said it equal to three are one plus. Negative or two not will. Give us this Major X right here. We&#x27;ll get zero six Negative. 13 this rural states. Same. And the last room will also save them. Okay, I know it will Dio is we&#x27;ll just go ahead and swap these two roads so it&#x27;s closer to reduce social reform. So 30 for three 06 negative. 13 00 negative. A squared plus two and a minus two. Okay, so we want no free variable. So what that means is we want to make sure that this entry is not equal to zero. Okay, So in order to that, we set a squared plus two and not the equal to zero We saw for a squared. When we get that a and not be equal to the spirit of two, and if that condition holds, that&#x27;ll make sure that there is exactly one solution and this is sort of operation. Okay, So onto the last word, I don&#x27;t want to find value for a for which there are infinitely many solutions. And this means that the reduce national inform has at least one free variable so greater than or equal to 13 variables. So by the previous implication that we&#x27;ve already done a few times, I&#x27;ll just right are reflex of broad matrix. So 30 for three 06 negative. 1300 negative. A squared was two and a minus two. And we just got that matrix from right here. That&#x27;s where we got it from. Okay, So if we want at least one free variable, that means that we want toe Look at this road right here. And we want to make sure that this is equal to zero and that this is equal to zero. So in order to do that, we sent negative a squared close to you. Well, zero. And we saw for that. And we get that, um well, so yeah, we want to sell for that. And then next up, we won&#x27;t solve Bernie minus two is equal to zero. So we saw this really quickly. We know that we want eight equal to and so we&#x27;re gonna plug this, eh Into this equation will see we have negative a squared plus two equals zero That gives this negative two squared was too equal to zero. And then this gives us negative four plus two has to be equal to zero. And then we send finest and see that negative too has toe equal to zero. But we know that that&#x27;s not true. So we know that this part must be wrong and that part&#x27;s not true, which means that there are no values for an infinite solution.

Rebecca Polasek · Answer

a system of three. The near equations in three variables that has a single order triple in its solution set is called On Independent system.

Problem 58 Medium Difficulty

Discussion
Make up a system of three linear equations in three variables for which the solution set is $\{(0,0,0)\} .$ A system with this solution set is called a homogeneous system. Why do you think it is given that name?

Answer

because constant term of each equation is zero.

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Mark Dugopolski

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Solving Absolute Value Inequalities

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because constant term of each equation is zero.

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