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Give an example of an inconsistent underdetermined systemof two equations in three unknowns.

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Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 2

Row Reduction and Echelon Forms

Introduction to Matrices

Missouri State University

Campbell University

Harvey Mudd College

Idaho State University

Lectures

01:32

In mathematics, the absolu…

01:11

01:25

Consider a system of three…

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Give an example of a syste…

01:08

Describe a method for writ…

02:38

What makes a system of thr…

01:17

State the conditions that …

00:36

Find a system of equations…

01:23

Write an inconsistent syst…

02:31

State three methods that c…

Describe a procedure that …

01:47

Give examples of a sys…

this question is asking us to solve the system, and it gives us some equations with A B and C, which we know are all constant. So our first up is going to be you to write out The Matrix from the equations that were given. So the first equation Reed's Ex one plus x two plus x three is equal to a The 2nd 1 reads to X one. Yeah, plus two x three is equal to be Finally, our last one reads three x to those three x three is equal to see. The first thing that we're going to want to do is reduce this major so that we can easily solve for X one x to an extreme. So a good first move would be to try and get rid of this entry and make it equal to zero. So we'll take our two and said it equal to negative to our one plus are too. Once we do that, we're going to get the falling matrix of first row, stays the same second row, become zero negative to zero, Then we have hey minus two a. Then our last road stays the same. Except we're gonna try and simplify this a little bit further. So first, we're gonna go ahead and which these two rows with each other and this is so we're gonna set or two well, to our three and our three equal to our two. And that's going to give us the following, Major. Okay, next up, we want to get rid of this term and make it equal to zero. So we're gonna set our three equal to 2/3 are, too. Plus our three that's gonna result in this matrix. Okay, so this is simplified enough to start solving for X one X to the next three. So let's go ahead and do that. I'm just gonna rewrite that matrix over here so we can see what we need to solve for our X one x you and extreme variables. Okay, so let's solve for X three. First double user, second row. When we see that we have two X three is equal. 2 2/3 c Must be bonus to a We're gonna divide both sides by two. We're going to see that we have extra A is equal to 1/3. Hey, what's 1/2 being minus a we're gonna go ahead and solve for X to using second bro. And that is going toe at first. Read three x two plus three x three is equal to see. So since we've already sold for X three, we can get a value for X two in terms of on Lee A's B's or sees, though it's rewrite, this is three x two plus three times 1/3 C plus 1/2 B minus a equal See? And I was just a substitution of this x three Valium. So we're gonna go ahead and simple find out we get reacts to worst E since these three ce cancel was 2/3 or three house Hey, minus three a. He will see. And then we're gonna go ahead and solve for X, too. So we have three X two is equal to three. A minus three house be says we can see moving that over to this side would make sees cancel. And finally we get that we have. X two is equal to a minus 1/2 a t. Okay, lastly, we're gonna go ahead and solve for X one using our first row, we see that we have x one plus x two was extreme equals ed and like before, we want this in terms of a, B and T. So we're gonna go ahead and plug in our ex to an X three values that we found here. Okay, so that's going to give us X one waas a minus one. Happy waas 1/3 C plus 1/2 B minus a Why So now we're gonna go ahead and simplify that you get exploring plus a minus one happy bus, 1/3 C plus 1/2 B minus A equals A. These days they're going to cancel out this. These terms negative one happened. What happened? They're gonna cancel out, and what we're left with is excellent is equal to a minus 1/3 c. So now if we want to write that in a vector notation we have, we can remind ourselves of our values. So we have extra A is equal to 1/3 C plus 1/2 being minus a. We have that X two is equal to a minus 1/2 B and then we have explored its equal thio a minus 1/3 c. So we write that that's one that's two. That's three is equal to a time. Some vector must be plus C, and we can start with our terms. So for X one, the coefficient of a is one b rex to also one and for experience negative one for B. Excellent doesn't have a B term, so we know that it was the equal zero for X to the coefficient is negative 1/2 and for X three, the coefficient is possible in half and lastly, mercy. We can see that for X one hour. See, coefficient is negative. 1/3 or ex to doesn't have a seat, herb. So we know that coefficient must be equal to zero. And for extreme, you can clearly see that it's equal to 1/3.

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