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Solve the following system of equation by Gauss-SeidelMethod a) 3p + 6y − z = 85x + y + 4p = 1106x + 15y + 2z = 72b) Using x0 = p, y0 = −11 + p, z0 = 3x...

Question

Solve the following system of equation by Gauss-SeidelMethod a) 3p + 6y − z = 85x + y + 4p = 1106x + 15y + 2z = 72b) Using x0 = p, y0 = −11 + p, z0 = 3x + 3y + 52z = 173.61x − 27y + 2z = 71.3141x − 2y + 3z = 65.46Note P = 4

solve the following system of equation by Gauss-Seidel Method a) 3p + 6y − z = 85 x + y + 4p = 110 6x + 15y + 2z = 72 b) Using x0 = p, y0 = −11 + p, z0 = 3 x + 3y + 52z = 173.61 x − 27y + 2z = 71.31 41x − 2y + 3z = 65.46 Note P = 4



Answers

Consider the following system: $$ \left\{\begin{array}{l} x+y+3 z=0 \\ x+2 y+5 z=0 \\ x-4 y-8 z=0 \end{array}\right. $$ (a) Without doing any calculations, find one obvious solution of this system. (b) Calculate the determinant $D$ (c) List all solutions of this system.

We need to simplify the given system of equations three X plus five way is equal to remind a +72 X plus seven said is equal Toby, too, and four y plus threes. That is equal. Toby minus eight. Now, by comparing it with a standard form we have, the value of everyone is equal. Toby. Three. The value of air to physical Toby to the value of eight raise equal to be zero. The value of Beaven physical Toby five. The value of B two is equal to zero. The value of B three is it called Toby, for the value of C one is equal to be zero. The value of C two is equal to be seven. The value of C three is equal to be three. The value of divan is equal to minus seven. The value of the U two is equal to be to the value of day three is equal to B minus it. We need to find the determinant Dave Weed is equal to, without the determinant of the metrics formed by the coefficients off the variables that is determinant of the matrix, with entries even a two a three b one b two b three c one c two c tree number Putting the values we have B is equal to bidder Determinant of the matrix with entries 3 to 0504073 Now we're simplifying. Determinant is comes out to be equal Toby minus 114 Now this is not equal to zero. So we can apply hair grammar rule for the solution that we need to find the value of DX, which is equal to a bidder. Determinant off the metrics with entries divan de toe de tree be even be toe B three c even see toe ct never putting the values we have the determinant off the metrics with entries is equal to B minus 750207 minus eight for three number simplifying determinant is comes out to be called to minus 140. Now we need to find the well of the way which is equal to widow determinant of the matrix with entries even a tow a tree divan de toe de tree, see, even see to city now putting the values. This is a call to bidder determinant of the matrix with entries three toe zero minus seven to minus 8073 Now we're simplifying. The dominant is comes out to be called toe to 28. Now we need to find the valley off. These ad, which is equal to bidder determinant off the metrics with entries, is equal to be even a tow a three. Be even be toe beat three d even de toe Petri. Now putting the values it becomes is equal to widder determinant off the metrics with entries 35 minus 7 to 0 to zero for minus it. Now we're simplifying. It becomes is equal to be zero. According to Kraemer rule, the value of X is equal to B D. X. Over the value of y is equal to B B by over. The value of said is equal to B D said over the never putting the values we have. The value of X is equal to B minus 114 be ready to buy minus 114 The value of Y is equal to be to 28 divided by R minus 114 and the value of said is equal. Toby zero, divided by 140 so this becomes the value of X is comes out to be called to one develop. Why is comes out to be cooled to minus two and the value of said is comes out to be called to zero, so this is over required solution.

Were asked to solve the falling system of X plus y plus Z is equal to 9000 and 0.5 x plus 0.6 Why plus 0.9 z is equal to 7 10 and that Z is equal to three y now. The first thing you want to notice is that this third equations equals 23 Why is kind of a gimme? That makes the question easier? Because we don't have three unique equations. We have Z equals three y, which we can immediately substitute into our other equations. To get that X plus y plus three y equals 9000 and 0.5 x plus 0.6 Why plus 0.27 uh, Z. Why is you go to 7 10 and we still have Zico's three wide. So now we have two equations in two variables, and this could be solved like, um, the same. We have three systems is just We have one less variable, which would make it easier on us. So now we need to solve fry their X or Y, and we can do this by using the first equation and saying an equal to X so doing this, I will give that X Plus four y is equal to 9000 and tracking the four My over will give The X is equal to 9000 minus four. Why, You know, we can substitute 9000 minus for Why? Into the equation. We got rights here to solve for what? Why is equal to so this will give 0.5 times nine K minus four wife plus 0.33 Why is equal to seven ton and solving for this will give 0.3 point 13 Why is equal to 7 10 minus 450 which will give that Why is you go to 2000 and because Z is equal to three y and we now know that why is he going 2000 we can immediately solve for Z Z will be equal to three times 2000. We should give a value of 6000. So now we have wise You go to 2000 and zeros of 6000. And like with all previous system equations, once we solve for two variables, we can plug both those rebels into any of the three systems that contain all three variables to solve the problem. So doing this with the previous system he came up with, um X plus four y equals 9000. We have X Plus four y is equal to 9000. So for times 2000 and 8000 which tells us that X is equal to 1000. So the ah solution to this particular system is X is equal to 1000. Why is you go to 2000 and Z is equal to sick?

All right. This time we have a pair of similar systems of equations, and we want to write out each determinant for the systems and decide if Cramer's rule is a possible option for solving the systems. So first things first. We want to right D for a party, and we do that by writing out of the coefficient matrix are given system. Why commence Just taking the coefficients of each variable in each of our equations in this system just like that, and we'll save that for later and continue writing out or determinants starting with a piece of X. And this is the same as our first determinant, but with the first column swapped out for our constant cone like this. All right, so now we've got Do you see why? Which again is swapping only one column this time. The middle column For the constant columns like so and finally, we have the subsea rich swaps out only the last column for the getting the hand. So for the constant column, all right, now that we have all those, we want to check and see if we can actually use Cramer's rule to solve this system and to do that. We solve our first determinant. And once we've got all that work to come out to a value of 22 and since 22 does not equal zero, Cramer's rule is usable for this system. All right, No, we'll move on to Part B, and we've got a whole different set of coefficients. So we've got for new matrices as films. All right? No, that we've got all those we just want to solve for D once again, all right, And once we add up, multiply through here we get the value of zero. And since we have a zero, that means we cannot use Cramer's rule for this particular system. And that's because our equations from Cramer's rule, for example, to solve for X we use these setbacks divided by D end of d zero. We just run into all kinds of problems dividing by zero. It is not

You have to write it in the cart. Explore Matt. So it's tube star ex minus y plus Z equals seven three Star X power to thus three star Why, it was Z thus three IHS star Why? Thus, to Star Z equals phone on putting it in. You get the solution. X equals 1.6, though line why was settled 0.9872 on Z, it was 2.794


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