5

0ozSPaC20+4 {4377e e/edd0+ 72v6-mMorK SPACY19 = /4Ll = ]The @ 'Pmston 4/28 + 0} AnarnWoak SPac0s+6 904 606BoSPKmal Gmesjemeen4r 17-%6rLm 7 '0 &'...

Question

0ozSPaC20+4 {4377e e/edd0+ 72v6-mMorK SPACY19 = /4Ll = ]The @ 'Pmston 4/28 + 0} AnarnWoak SPac0s+6 904 606BoSPKmal Gmesjemeen4r 17-%6rLm 7 '0 &' 44ELAdSt"_=MWnal647equatoto teCeratornWoak SRpant (-2,-8)au1ras sd7yeg - 2DTej -77eJ-14,-}-2BONUS: WuST @ITCOMPLETELY CorcT. Worth2pontSIIISolve torx: 2(5-4)2 {(-1)

0ozSPaC 20+4 {43 77e e/edd0+ 72v6-m MorK SPAC Y 19 = /4 Ll = ] The @ 'Pmston 4/28 + 0} Anarn Woak SPac 0s+6 904 6 06 BoSP Kmal G mesjemeen4r 17-%6r Lm 7 ' 0 &' 44 ELAd St"_=M Wnal647equatoto te Ceratorn Woak SR pant (-2,-8)au1ras sd7 yeg - 2 DTej -7 7eJ-1 4,-}-2 BONUS: WuST @ITCOMPLETELY CorcT. Worth2pontSIII Solve torx: 2(5-4)2 {(-1)



Answers

If $\mathrm{f}=\{(2,4),(3,6),(4,8)\}$ and $\mathrm{g}=\{(4,3),(6,4),(8,2)\}$, then find fo g. (1) $\{(2,3),(3,4),(4,2)\}$ (2) $\{(4,6),(6,8),(8,4)\}$ (3) $\{(3,2),(4,3),(2,4)\}$ (4) $\{(6,4),(8,6),(4,8)\}$

All right. So we're gonna use are inverse to find our solution. And we already know the in vs because they're from the previous section. So just looking this up a would be negative for 16 Negative too. Our X would be our variables. X and Y B is our solution, says 0 14 So we have to find the inverse for a so negative for 16 negative to I have them all written on my next page since we've already solved them all. Negative. 416 negative too. Her in verses. Negative. One negative. 1/2. They have three native to three names, too. So all I need to do now is take that matrix I just found and multiply it by our solutions. So negative. One times zero plus negative. 1/2 times 14 I get it three times zero plus negative. Two times 14. So NATO one times zero is zero negative. 1/2 of 14 is negative. Seven negative. Three times zero is zero. Knew of two times 14 is negative. 28. So that tells us X is negative. Seven. Why is negative? 28. So our solution would be native seven Negative. 28

Just Hello. So here are given um these small equations negative four X plus y equals zero and six X minus two. Y equals 14. So therefore we can have our matrix A. B just equal to the matrix of the coefficient on our variables. So is equal to negative +416 negative two. Then we get the system of equations. We get that a times the column vector X. Y is well equal to the solutions, which is going to the constants, which is going to be equal to the column vector 0 14. Yeah, Okay. Or in other words, we get that X. Y. So from this we get that our column vector X Y is going to be equal to well, a inverse right now, playing most of my universe, this is equal to a inverse times the column vector 0 14. So we found the common because we found a inverse in a proceeding um problem, right, A inverse. We just concatenate a with the two by two identity. And then used elementary operations right to get the identity of left hand side. What's on the right hand side is a inverse. We find that a inverse is going to be equal to on the matrix matrix negative one, negative one half negative three. Um negative too. Okay. Um and then well we want that xy again is equal to a inverse times the column vector 0 14. So therefore we get that X y is going to be equal to a inverse, which is negative one negative one half negative three negative two. Um multiplying by um by 0 14. So again here right we have a two by two matrix multiplying by a two by one matrix we get to buy one matrix to go ahead and do the dot product here. So our first entry is going to be first row times the first column, the dot product. So negative one times zero plus negative one half times 14. That gives us a negative seven. And then um dot product the other way from the second row. Um here with the column is going to be negative three times zero plus negative two times 14, which is a negative 28. So therefore we get our column vector here, xy as negative seven, negative 28. Therefore we get that the value of X is going to be equal to negative seven and that Y is equal to negative 28. Yeah. All right. Take care

I wouldn't use Cramer's rule to solve the system of equations. Using Cramer's rule, would first want to evaluate the determinant of the coefficient matrix? Yeah, yeah. Using a calculator to multiply and subtract across products and you end up getting To to the Gospel Places .37. Yeah. No, by Cramer's rule says that X is equal to the determinant of the matrix formed by constance and the coefficients are Y over our to determine how the coefficient matrix. And We'll say that's 1851.36 Over .37 And 1851.36 over .37 is like 5003.6, 8. Okay. Why is equal to the determinant of the matrix formed by the coefficients on the X. And the constants over deep? And the determinant Is 1876.63 and so divided by a point 37 We could have 5,071.97. Okay.

All right. So we're gonna use the inverse of a matrix to solve this, knowing that when we have it set up its original equation, we take our answer multiple for the inverse we get thes correct solution. So, for example, if it was three negative one native to one, so our equations R b would be our solutions, which are 84 Our X is our variables. So x and why and are in verse, which I have written down from all of our previous problems. So three negative one negative to one has an inverse of 1123 1123 So we're gonna do is we're gonna multiply are in verse by our solution I'm not should get us our answer So one times eight plus one tons for and two times H plus three times for so eight plus four is 12 and 16 plus 12 28. So that tells us X is 12 And why is 28? So our solution b 12 28


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