This question is asking us to solve the given system of linear equations based on the reduced matrix out we have been given. So well, go ahead and first label our columns with the variables that the representing. So this is X one. This is X two, this is X three. So for the 1st 1 we can write out the equation with the coefficients for X one x two x three has given to us by this matrix. So the coefficient of X one is one. So we can just leave that out since it's a one. Or we could put that in if we want. And then the coefficient of X to an X three are both zero. So we can basically just ignore these and write X one equals negative three. Because these cancel out one times X one is just one X one. So we get that X one is equal to negative three. Uh, this same thing we're gonna bring you the same thing. For rose to one rose three. We have zero x one plus one. Next to zero x three is equal to zero. We simplify this, we get X two is equal to zero. Kathy's cancel out and one times X two is just x two. So lastly, will Sof uh, this final road here? So we have zero x one plus zero x two was one x three is equal to seven. We simplify that we get that extra is equal to seven so we can write our final answer in Dr notation. If we want, we can say this x factor of these three components x one x two x three is equal to negative three zero seven. Okay, so we're going to do the same thing for this. We'll label our variables, x one x two x three x four and also, uh, just tow. Remind ourselves this lost column is what the equations in this matrix are equal to. So for the first row B right x one zero x two plus x three plus zero x three was negative. Seven x four is equal to a and what we want to do before we solve. This is looking or matrix and see if any of these variables our free variables and what that means is that one of these variables x one x two of three or X four is not a leading injury. So here we see that this is a leading entry. This is a leading entry and this is weeding entry, which means that this is not and none of the entries in the X four column are reading injuries. So that means, uh, export is our free variable. And we want to solve all of our equations that we're gonna write down. We're gonna want to solve for X one, x two and x three in terms of x four. So we'll go ahead and do that. So he's cancel because these are both multiplied by zeros. We're left with X one minus seven. X four is equal to a And like we said before, scores are free variable. So we want to solve for X one in terms of export. So x one is equal to eight plus seven x for Okay, so now we'll go ahead and look at our second row. We have zero x one plus x +20 x three plus three of four is equal to two. He's cancel. We have x two plus three x four is equal to two and we solve for x two in terms of our free variable of exports, so actually was equal to two minus three x four. So then our last row, we're gonna want to solve for X three. So we have zero x one. The zero x two was x three plus explore is equal to negative five. He's cancel. We solve for X three in terms of explore, So x three is equal to negative five. Linus Export. Now what we're going to want to do is do what we did in the first problem and write that in vector notation. So we have the general expect, er that we're selling for which has components excellent X two and X three. And what we're going to do is we're going to May one. Dr. That's just these numbers that have no other variables involved. So we see our X one has a next to has it too. Exterior has the negative five. We're going to add that to explore times it's coefficient, so we can see that the coefficient for X one for X four, And, uh, this overall equation for X one is seven. So we're gonna write a seven here, uh, for X to the coefficient of X four is negative three. And for X three, it is negative one. And if this is looking a little bit confusing to you, we'll just go over here and know that we can rewrite this as we can. Reword right each of these lines the x one x two x three as x one It wa ce eight waas x four times seven, which is seven. Explore and we can see that that is exactly what we got here. We can do the same thing out for Ex tube just for practice. So x two is equal tune too. Shot number here. Plus that's four times negative three, which is negative three x four, which is the exact same thing that we have here. And lastly, X three is equal to negative five plus negative one x four, which is also the same thing that we see here, okay, onto the next one. So we'll do what we've been doing right down the different variables in these systems of equations. We have excellent Xbox three x four, index five and we then want to figure out what our free variables are, and we can see that This is a leading entry. This is leading entry. This is leading country. So that leaves next to an X five to be our free variables because they have no leading injuries. So we'll solve this just for myself. The other ones, uh, we'll start with the first column right here. You see that? This is X one minus six, x two plus three x five is equal to negative too. And remember, we don't have an extra year and explore because these are both coefficients of zero, which cancels he's out. So then we want to solve for X one in terms of our free variables, which our ex too and x five So we have X one is equal to negative two plus six x two was er sorry no plus minus three x five and then we'll do the same thing for the next row where we have, uh zero x one plus zero x two plus x three closed zero x four plus or expire is equal to seven. We want to solve in terms of our free variables which we have his x five and we get the, uh X three is equal to seven minus for X five. Um, and now we have our last row that we want to D'oh! And so this one is going to be solving for X four. So we are going to I'm gonna open up a other patriot. Quick. Um, so we're going to have X four five x five is equal to eight, and then we want to solve it in terms of our free variable, which is that's five, and we have explore is equal to eight minus five ex spies. So now we have the variable. So we want to solve for which is excellent x three, next four over leading injuries. We'll write down what we got for those. So we got X one is equal toe. Uh, negative two plus six x Q minus three x five. Well, go ahead and write that down for extremely. We know that that equals seven minus four x five. Then for last variable, which is explore, we have a minus five x five. If you want to put it in vector notation like we did with the other ones, we write x one. It's three that's for is equal to thinking of 27 a Waas X two, which is one of our free Barry Bulls time. Some doctor Waas That's five times some vector. So now let's go ahead and fill that in for X two. We see that X one has a coefficient of six. Since there's no X two terms in either extra explore, we know that these have a coefficient of zero because that'll cancel it out. And for X five, we can see the coefficients for X one. It's negative. Three for X two. It's negative for and for explore its negative five. Okay, so now we'll go ahead and go into the last one. So for this, before we even start solving for variables, we can kind of do a little shortcut and just take a look at this bottom row. So what this bottom row is saying is that zero x one. I'll label the columns again for the variables we have x one x two, the next three. It's saying zero x one plus zero x two zero Expiry is equal to one. So because he's a zeros zero times, any number is gonna be zero. So we have zero plus zero plus zero is equal to negative one. Er sorry is equal to one. And some of all zeros is going to Syria. We have zero equals one, which is obviously not true. So for this matrix because we have this which is invalid, we say that this matrix has no solution.