write-the-difference-equations-in-exercises-29-and-30-as-first-order-systems-mathbfx_k1a-mathbfx_k-f

Question

Write the difference equations in Exercises 29 and 30 as first-order systems, $\mathbf{x}_{k+1}=A \mathbf{x}_{k},$ for all $k$ 
$$
y_{k+4}-6 y_{k+3}+8 y_{k+2}+6 y_{k+1}-9 y_{k}=0
$$

Michael Jacobsen · Accepted Answer

in this example, we have 1/4 order difference equation that&#x27;s provided here, and our goal is to do the following. So our goal is to write this particular equation in the form of a vector. X K plus one equals a particular matrix, a times except K. So if we can determine what the Matrix A should be and what the vector except cave must be, then our work is complete. Let&#x27;s do some early analysis on this. This particular equation is a Order four difference equation. That means our vector X K is going to have four elements all together. Next, let&#x27;s use a theorem that&#x27;s also going to guide our work. So the theory tells us if we have a difference equation of order N, then we&#x27;ll have a vector x k that&#x27;s formed in this way with four entries altogether, we start with y que then increase on the sub scripts on K. So the second entry is why K plus one. Then why K plus two? Finally, why K plus three. Now we&#x27;re ready to describe the Matrix, and if we have a order, N and N is equal to four r matrix is going to be of size four by four. Let&#x27;s make it appropriate size for that task. And what the rose air telling us in this theorem is will start with a zero will produce a one and then the remaining entries are all zeros. So I have 0100 as the first row of this matrix. Now, the theory tells us that the only difference amongst the Rose is that if there was a pivot here, the prove, it moves down to the next column and then we&#x27;ll have zeros thereafter. So back to a matrix say will go 00 place or pivot here and places zero next. Next, go 000 and place the pivot here. And that&#x27;s our 1st 3 rows. Now the next key part are these quantities that we see here in the final row. Notice the sub scripting the sub scripting tells us that will have in our case and a one to be negative. Nine. A two is six. A three is eight and a four is equal to six. So oh dear, rather than the sub scripting before our A one is negative. Six. A two is eight 83 6 and a four is negative nine. So don&#x27;t make that mistake. The sub scripting increases from left to right, starting with E K plus three for the Order four Difference Equation. Then we go in the reverse order, starting with a four. Take its opposite. So we need the opposite of this quantity, which makes it a positive nine produced here. Then the very next one. We take the opposite of this value, so we&#x27;ll have a negative six. Likewise, take the negative off A two for a negative eight. The negative off a one gives us a positive six. And now we have produced the required matrix. So with this matrix given here with XK provided here, we now have this particular equation which captures the same information as the original difference equation.

Nikolaos Alexandrakis · Answer

we will right this third or their difference occasion as the first or the system in the form X killed plus one. We will write it with over line just to know that this is a vector. It was a times ex gay Which way is three is three times three mavericks Since we&#x27;ll have the thunder, their system and we proceed by considering the vector x t Uh, it&#x27;s consistent these three values and the vector x k plus one which is this vector for following the definition off ex gay and we write Get with the foot taking the talk on the information from thing this and get some of them got played Tow the federal court in areas see off Gabe Plus three equals three over four CK plus two minus one over 16 ck and we perceive as in the book. And, uh, that&#x27;s an example of the book. And we write it as they seduce them in the for month eight times ck where a equals one or one 01 and minus one over. So extreme. Oh, and city over four. All of these times, the vector X Games

Nikolaos Alexandrakis · Answer

way. Want to look? Find the order off this discreet regulation. Now, as we know from theory, the order off the discredit creation is the difference off the highest you dicks to the lowest index. But any three is here is a perimeter. So if it is no zero, then we immediately have that. The order is three, cause we have K plus three minus G sequence three. Now, if a three they&#x27;re called zero Oh, just to fix this three, then we proceed with a difference off. You&#x27;re a blustery and your plus one. And the difference is too if a two is not zero because okay, plus reminders. Okay, less one that close. Go now if a door he&#x27;s zero. Then again, the difference with the order of vacation will be the difference Off the fire started their toe toe the next. So the first Norman reveal or their, uh their toe to the order off the third off the first non trivial term. Then if a one is not zero, then the corresponding or there is one. Otherwise, we have the correspondent thing or there is zero minutes. Otherwise, we have why gay plus three queens zero with which would later toe like a a quint zero, which is review

Supratim Roy · Answer

Hello, everyone. So here we have a system offline. Any questions off to Perryville&#x27;s And we&#x27;re to find a solution by using any convenient method. Okay, so you can see there are three questions where we can eliminate why, Okay, So if we have to eliminate why, then we have to make the coefficient off. Why equal in both of the questions and then subtract one from the other? Okay, so to make the coefficient of y on the second equation three, we have to multiply the second equation by three. Okay. And then subtract the second big ocean from the first equation. So if I name it A on this one, be so we have to perform this operation a minor&#x27;s three be OK, So is two X plus three y equals 2 10 on the question be multiplied with three would be so minus three x multiplied with trees negative nine x Then why multiplied with three would be three way equals two three multiplied with 18 would give us 30. Sorry. 54. Ok, so now we will subtract it. So we have to use subject. So this Simon changed to positive This one would change too negative. And this one So negative. Okay, So positive minus at it to try sex would give you 11 x and then three way and negative too. I would get cancelled. 20 and positive. 10 negative. 54 would give negative 44. OK, so that means we will have 11 x equals negative 44. So axl b equals to nearly 44. Divided by 11 which is it goes to negative four. OK, so we have found X. Now, we will use this value off X on the second equation to find the value of y. Okay, so let me right here. The second equation is negative. Three x plus y. It was to 18. Fine. So we will have negative three multiplied with negative four plus y equals 2 18 Okay, so that would give us positive. 12 plus y equals 2 18 So why would be close to 18 minus 12 which is a course to six. So the will of white six. So here we can write the solution in terms off ordered Baird, which is nearly full comma six. And since this is the only solution for the system off equation, so then their solutions said, is this one Okay, so I hope you have understood. Thank you.

Melissa Stefan · Answer

let&#x27;s use the substitution method to solve the system. Start by solving the first equation for why to get why by itself will divide both sides of the equation by three. This gives us why equals eight minus three X. We can then substitute this expression eight minus three X into our second equation for why three X plus eight minus three. Next is equal to eight. When we combine our like terms, notice our X variable is going to cancel out and we&#x27;re left with eight equals eight. This is a true statement, so we can conclude that both of these lines have the exact same solution set. We can write that solution set as every ordered pair x y, where three x plus y equals eight. There are infinitely many solutions.

Melissa Stefan · Answer

we&#x27;re going to use the elimination method to solve this system. But we&#x27;re gonna do a little bit different because of all the decimals. I think this is gonna produce some really funky numbers. So we&#x27;re gonna do this very carefully and eliminate each variable separately to come to our final answer. So let&#x27;s start by eliminating our X variable first. And to do this we want to make sure that those coefficients are opposites so we can multiply our first equation by 1.7 and our second equation by negative 3.6. This is going to result in a new system that looks like 6.12 x plus 4.93 Why is equal to 40.307 and our second equation is going to be negative 6.12 X plus 16.2. Why is equal to negative 52.704 Now, when we add these two equations together, we end up with the ex terms eliminated, leaving us with 21.13 y is equal to 12.397 Now we can get rid of our decimals at this point. If we multiply our equation by 1000. This will leave us with 21130 y is equal to 12 397 Now. To do this, I I multiplied by 1000 to get rid of the decimal so that we don&#x27;t have a a decimal that goes on forever, or an irrational number that we can&#x27;t easily write as a fraction. By getting rid of the decimals, we can then divide both sides by 123,130 which results in a fraction that cannot be simplified. So we have a value of y equals 12,397 divided by 21,130. Now, if we wanted to round this to the nearest 100 or 10th we would go ahead and use a calculator. And why would be approximately zero point six? We&#x27;re not gonna use this value to substitute to find X, because it&#x27;s approximated, and it&#x27;s not usually a good idea to approximate values. Instead, we&#x27;re going to go back to our original system and then eliminate our why variable to solve for X. So if we use our original equation 3.6 x plus 2.9 y equals 23.71 and 1.7 x minus 4.5 y equals 14 0.64 to eliminate the why we&#x27;re going to multiply the first equation by 4.5 and the second equation by 2.9. Now we&#x27;ll have new equations scroll down for some space of 16.2 x plus 13.5 Why is equal to 106.695 crazy decimals again? And our second equation is 4.93 x minus 13.5 Why is equal to 42.456 now with our Y variables having opposite coefficients? When we add these two together we end up with 21.13 X is equal to 149.151 and again, we&#x27;re going to use that same trick is before and multiply the entire result here by 1000. To eliminate those decimals, we can keep our answers as exact as possible, so multiply this by 1000 and we have 21,130 x is equal to 14 91 51. So 149,151 crazy big number divide will side by 21 1 30 And we have X is equal to 149,151 divided by 21,130 Again, that would be your exact value if we wanted to round this to the nearest 10th go ahead and use a calculator and we&#x27;d say X is approximately 7.1. So the approximate solution for this system is the 0.7 point one zero point things.

Problem

Write the difference equations in Exercises 29 an…

Problem 29 Hard Difficulty

Write the difference equations in Exercises 29 and 30 as first-order systems, $\mathbf{x}_{k+1}=A \mathbf{x}_{k},$ for all $k$
$$
y_{k+4}-6 y_{k+3}+8 y_{k+2}+6 y_{k+1}-9 y_{k}=0
$$

Answer

$x _ { k + 1 } = A x _ { k }$

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

Linear Algebra and Its Applications

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$x _ { k + 1 } = A x _ { k }$

Linear Algebra and Its Applications

Lily A.

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

Lily A.

Heather Z.

Kristen K.

Joseph L.

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

Linear Algebra and Its Applications