describe-and-compare-the-solution-sets-of-x_1-3-x_25-x_30-and-x_1-3-x_25-x_34

Name: describe and compare the solution sets of x_1 3 x_25 x_30 and x_1 3 x_25 x_34
Uploaded: 2019-07-29
Duration: 4 min 13 s
Channel: Numerade
Description: Describe and compare the solution sets of $x_{1}-3 x_{2}+5 x_{3}=0$ and $x_{1}-3 x_{2}+5 x_{3}=4$

Question

Describe and compare the solution sets of $x_{1}-3 x_{2}+5 x_{3}=0$ and $x_{1}-3 x_{2}+5 x_{3}=4$

Runpeng Li · Accepted Answer

all right. So in these problem, we were given two equations and that we need to compare the solution. Says how these two equations. So we first have x one, minus three, x two thanks to plus five x three, which is zero. So we can who who tow we can put x x two terms and x three terms. Um, on the right hand side. So that is X one because the X two minus five x three So that means we have a social set that x one determined by X to the next three and extra and excrete are three variables here. That&#x27;s three x two minus five x three and x to the next three. Now we separate these factors by, uh bye bye. The terms that contain X to annex three. So we first to check out the terms of containing X juice with three ex too x two and zero plus next five x three zero and next three. So that is our first solution. You compare Measure the vector for so that is 310 time sex too. And next, five 01 times x three. All right, So for their first equal for the second equation. Here we have X one minus three x two I was five x three equals four. Now again, we put a three x two and five x three on the right hand side, so that is X one can be returned. Informs off ex to a next three. So that&#x27;s three x two minus five x three plus four. So this ocean said this three x to minus five x three plus four and next to x three are three variables. Now we separate this vector that is three x two x 20 and active five x three X That should be zero and next three last term. He&#x27;s accounts tent, which is for you to do now we take out a skater, so I will Right down here. Now we take out a skater. So that is three one zoo times x two and next five 01 times x three and the council in term is or 00 So that that is our solution for the 2nd 1 So we could compare these two solutions. We can actually find out that the only difference is that the second solution has accounts in term here that it&#x27;s because because we have Ah, we have, Ah, homogeneous least time here. So we have a council for my hand side up the equality. But Porter Percy, for the first equation, we don&#x27;t have such a such a council way only have zero. So that is a homogeneous in your system. So we can running down as, ah, as a parametric direct victor forms without any constant better. But the second equation things are different. So that&#x27;s something we need to be careful.

Michael Jacobsen · Answer

in this example, we&#x27;re going to be handling two equations simultaneously where the only difference is between the equations are the right hand sides. So the first question equation is a homogeneous equation. Let&#x27;s handle this one first weaken. Begin by solving for X one in this equation so that X one is equal to negative. Nine x two plus four x three. Now we don&#x27;t know a lot of information about the variables extra next three. Other than that, they&#x27;re free variables in this situation. So all right X two is free as well as X three is free. Next, we can now describe the Vector X, which will be a solution to this system. In Parametric Vector form will be of the format x one, x two and x three, and we know the X one. If we solved it as we did here, is equal to negative nine next to plus four x three. Now Ex, too, because it&#x27;s a free variable has no restrictions. So we just place it here, and we&#x27;ve effectively told everyone reading the X two equals x two. It&#x27;s not a whole lot of information, but that does capture the essence of X to being a free variable. Likewise, X three is equal to x three. So a place it here then we can now say that X is the following. We take the free variables in this system x two x three and first record them. So put it next to here plus an X three here and now we&#x27;re going to put in the constant vectors. We get the constant vectors by looking at the coefficients first for ec two x two, which are negative 91 and zero, then for X three, the coefficients are 40 and one. We recognize that this equation or this vector form, is the quick equation for a plane through the origin or the zero vector in our three. It also must contain these two constant vectors as well. So you could use this content constant vectors to sketch this plane. Well, now that we know that this is the equation for a plane containing these vectors and going through this, you&#x27;re a vector. The question now turns to What is this really The only difference? Recall is the negative to here, So it&#x27;s copy our process. We can say that like as before x one is negative nine x two Mice four x three x two is free and x three is free That allows us to write X equals as we did before we could write out all this information. But the only difference is we have a negative to hear hope. I forgot to copy that negative to here as well. When we solve for X one so we&#x27;ll have that X. The solution is negative. Nine Next to plus four x three minus two then x two goes here since it&#x27;s free and x three will go here since it&#x27;s also a free variable. Then in Parametric Vector form, we can say that X is equal to first a copy x two Ah constant vector as we did before, plus X three times another constant vector, but will also add purely a constant vector with no scaler multiplication which is negative to zero and zero and that handles this portion then the rest of our work is purely identical. Saw copy. Negative. What? 910 and 401 So now we can say what this is effectively going to be geometrically So this is ah, plane that is obtained from X one plus nine. Next to mice four x three equals zero or, in other words, the previous playing we&#x27;re dealing with. So we take this plane and shift lift two units in the X one direction, which corresponds to the constant vector here telling us. Move the plane according to that vector. So that&#x27;s the difference between handling and equation that&#x27;s homogeneous versus one with the constant on the right hand side.

Amy Jiang · Answer

Okay, so first note that ex cannot be acquitted. Three. And now let&#x27;s multiply through by X minus three. So we get three X minus nine plus X plus one is equal to two ex money six and then minus five minus three. Actually, that&#x27;s plus three X. Okay, so we get four. X minus eight is equal to five X minus 11. Okay, so we let subject, for example, sides to get negative eight is equal to one X minus 11 and adding 11 on both sides. We gotta love in my eighth that use me. X is equal to three. And we just said that ex cannot be able to dream or it&#x27;s undefined. So our solutions that is empty.

Amy Jiang · Answer

Okay, So notice that we have a denominator of X minus one. So our solution cannot contain one or we get an undefined value. So let&#x27;s start by multiplying by X minus one, which is our LCD. So we get three X minus five is equal to tomb times X minus one one is to act. This is equal to two X minus to minus two X or two X&#x27;s cancel. So we get three act minus five. Poor at file. Did I decide to get three is equal to drink, so X is equal to one. So that&#x27;s that. We just said that ex cannot be equal to one or we get an undefined value, so we actually have no solutions.

Jingyun Wang · Answer

the crusher asked to determine a solution off the give a system. First, we need to turn the system off equation into a semantic metrics form. So take the core official out. Remember, when you do find the reduce road rational informed to find an answer. So for roll one, we can do throw one minus wrote you copy other rose for road to we can do to times roll one. My doesn&#x27;t wrote you for rosary. We can do five times through one by this roastery Copyright one for rosary. We can do nine times the road to minus roastery. Copy other rose for a rosary. We can do a negative one over. Study four times a rosary. Copy other rose Furrow to we Can Do Road to Plus five times and roastery. Yeah, copy other rose for a one we can do roll one plus two times 03 Copy Other rose You for no one we can do roll one minus row. Two. Copy other rose. So the final answer is excellent. Equal to zero ax to you Call to zero and xer it. Also, you could do their own

Jingyun Wang · Answer

the question asked to determine the solution off the system First, when you change a system off equation into a semantic a matrix form, so take the corporation out. Remember, we need to find a reduce row. Actually informed your funny answer. We have one here so we can switch your grocery with no one copy wrote to Furrow to We can do five times 01 minus wrote you in for grocery. We can do a true times roll one minus rosary Copy Row one to the elementary rule operation for road to on the Rosary For a rosary, we can do zero to minus two times Roastery Copy other rose for road to We can do 1/6 times zero to copy other rose every time when we see all zeros in one row, which means stairs of free variable. For this case, X ray is the favorable. It can be any number x two equal negative X ray x one equal negative for a Times X three minus X two according attacks, too, and you get X one equal to negative X ray and this is your final answer

Problem

In Exercises 19 and $20,$ find the parametric equ…

Problem 18 Hard Difficulty

Describe and compare the solution sets of $x_{1}-3 x_{2}+5 x_{3}=0$ and $x_{1}-3 x_{2}+5 x_{3}=4$

Answer

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

Anna Marie V.

Heather Z.

Kristen K.

Michael J.

Absolute Value - Example 1

Absolute Value - Example 2

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David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Anna Marie V.

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications