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(a) Solve the following system Of linear equations by reducing the augmented matrix of the system row echelon for_ Describe all the solutions of the systemX1 +3xz +...

Question

(a) Solve the following system Of linear equations by reducing the augmented matrix of the system row echelon for_ Describe all the solutions of the systemX1 +3xz + 2x3 =1 2x] + 8x2 + 6x3 X1 +2xz +*3 = -]-2-]is the augmented matrix of a system of linear equations in fourvariables X, Y; 2 W; describe the solution set of the system:

(a) Solve the following system Of linear equations by reducing the augmented matrix of the system row echelon for_ Describe all the solutions of the system X1 +3xz + 2x3 =1 2x] + 8x2 + 6x3 X1 +2xz +*3 = -] -2 -] is the augmented matrix of a system of linear equations in four variables X, Y; 2 W; describe the solution set of the system:



Answers

Solve the system of equations by finding the reduced row echelon form for the augmented matrix.
$$\begin{aligned} x+y-3 z &=1 \\ x-z-w &=2 \\ 2 x+y-4 z-w &=3 \end{aligned}$$

Here we have a system of three equations and we're going to solve the system using reduced row echelon form. So the first thing we want to do is convert the equations into an augmented matrix. So from the first equation, we have one x zero. Why negative one Z and two. For the second equation, we have negative to X one y three Z and negative five. And for the third equation, we have two x one y negative one z and three. That's our augmented matrix. And we are allowed to use a calculator to solve this problem. So I'm going to type that matrix into my calculator and then ask the calculator to find the reduced row echelon form. Okay, so once you have your calculator, you want to go into the mate Matrix menu over to edit and select matrix A. And then you're going to first type in the dimensions of the matrix three by four and then type in all the numbers. As you can see, I've already done. Once you have an empress and you go back to your home screen, then you go back into the Matrix menu over to math and scroll down until you find R R E F, which is reduced row echelon form. Select that and then back into the Matrix menu and select matrix a press enter. Okay, so here's the reduced row echelon form of AR augmented matrix, and we're going to write that down and interpret the answer from that. Okay, so let's take a look at what we have here. They're different things that can happen with reduced row echelon form leading you to ah, your interpretation of the answer. Typically, if there's just one solution, you're going to get something like this. If this happens with one's on the main diagonal and zeros elsewhere with a column of numbers A, B and C that tells you X equals number, eh? Why equals B and Zeke will see. But that's not what happened. We see that we got a row of all zeros, and that's an indication that we're going to have infinitely many answers. Sometimes we just say infinitely many solutions, but sometimes we can describe those solutions in a little bit more detail. So let's change row one and row to back into equations. So looking at Row One, we have X minus Z equals two. Now we could write that as X equals E plus two and looking at Row Two, we have why, plus Z equals negative one. And we could write that as why equals the opposite of Z minus one. So we have infinitely many answers, but they have to have these relationships between the values of X, y and Z.

Here we have a system of equations and we're going to solve for the solution. And it's really helpful if we stop, stop and think about this conceptually before we just start doing calculations. So each equation represents a line, and we're looking for where the lines intersect. And if you look at equations one and three, you'll notice that they're actually the same line. So we have a bit of redundancy here, So really, we're just looking for the intersection of equation one. Any equation, too? Now, even if you hadn't thought about that, you can still go through the procedure and this will work out fine. So the procedure is that we're going to first make the augmented matrix. So from equation won, we have won one and three, and from equation to we have to three and eight. And from Equation three, we have 22 and six. Now, we are allowed to use a calculator on this problem. So I would grab my calculator, type in the Matrix, and then have the calculator find the reduce row echelon form. Okay, so once you have your calculator, you're going to go into the matrix menu and then over to edit and select matrix A and then type in the dimensions three by three and type in all of the numbers. As you see I've already done. And then you can go back to your home screen and then back into the Matrix menu over to math and scroll down until you find R R E F, which is reduced row echelon form, press enter and then back into the matrix menu Select Matrix A and press enter. And this is our reduced row echelon form. So we're gonna write it down and then interpret our results from it. So you might notice that the third row is just zero equals zero. And that happened because we had that additional third equation that was actually equivalent to the first equation. So if we focus on the first and second rose of our Matrix and convert them back into equations, we have X equals one, and we have why equals two. So our answer to the system is the 0.12

So for this problem, we have a system of equations and noticed that there are only three equations. But there are four unknowns. So when we saw the system, the best we're going to get is a relationship between the variables. We're not going to get just a single new miracle answer. The first thing we want to do is convert our system of equations into an augmented matrix. So for the first equation, we have one x negative one. Why negative ones E to W and negative three. For the second equation, we have two X negative one. Why? Negative to Z three W and negative three. And for the third equation, we have one x negative, too. Why negative ones? See www and negative six. Now that we have that, we can use a calculator to find the reduced row echelon form. So let's grab a calculator and then you're going to go into the Matrix menu over to edit Select Matrix A. And you're gonna type in the dimensions of the matrix and then all the numbers in the Matrix, as you see I've already done. And then we're going to go back to the home screen. and then back into the Matrix menu over to math and scroll down until you find R R E F. Which stands for reduced Rochel on form. Select that and then back into the Matrix menu Select Matrix a impress Inter and this is our reduced through echelon form. So we're going to write it down and then figure out what it's telling us about our answer. Okay, so here's our reduced row echelon form, and if we take the first row, encumbered it back into an equation. We have X minus Z plus w equals zero, and we could write that as X equals Z minus w. And then for the second equation, the second row. If we convert it to an equation as well, we have y minus w equals three. So we could convert that into why equals W plus three. So those are the relationships between our variables that represent all the solutions to this system

In this problem. We have a system of three equations and we're going to solve the system using reduced row echelon form. So the first thing we want to do is convert the equations into an augmented matrix. So in the first row will have 1132 Those were the coefficients from the first equation and the constant. And in the second row will have 34 10 5 coefficients and constant from the second equation. And 1243 from the third equation. Okay, we are allowed to use our calculators to solve this problem. So what we're going to do is put this matrix into the calculator and then ask the calculator to compute the reduced row echelon form for us. So using your calculator, you can go into the matrix menu over to edit and select matrix A and then you type in the dimensions three by four I've already done. Then you go back to the home screen. Then you go back into the Matrix menu over to math and scroll down until you find r r e f, which stands for reduced row echelon form. Select that and then back into the Matrix new to select Matrix A and here's what we get. So then let's copy that down and interpret our answer from that. Okay, so here's our reduced row echelon form and you'll notice the third equation 0001 There's a contradiction there. That's like saying that zero Z is equal to one. Um, from that we can tell there is no solution to this system. If there had been a solution, what we would have expected to see is something like 100010001 with some numbers here. If that was the case, we would have one solution. There are other times when the third row has all four zeros in it. If that was the case, we would have infinitely many solutions, however, with what we got, because zero can't equal one that indicates no solution to this problem, so we can write no solution with the empty set symbol


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