WIR Math 166-copyright Joe Kahlig, 10A Page 1Week in Review # 8Section 4.3 and 4.4: Systems of linear equations.• Setting up word pro blems.• Define the variables. • Gauss-Jordan row operations.• Row-Reduced form (Reduced Row Echelon Form)• The first non zero number in a row is a one. Called a leading o ne.• The leading one is the only non-zero entry in its column.• The leading ones ar e positioned in a diagonal- like manner starting at the upper leftgoing to the lower right.• Types of solutions• no solutions when a row of the matrix gives a non-true statement, i.e. 0 = 5.• exact solution (when in row-reduced form)• number of leading ones = numb er of variables• infinite solution (when in row-reduced form)• number of leading ones < number of variables1. Set up the system of equations. Do not solve.Seven hundred tickets were sold for a performance of a play. The tickets cost $6 forseniors, $8 for adults, and $3.5 for children. The total receipt for the perfo r mance was$3,512.50. If three times as many children as a dults attend the play, how many of eachticket were sold?2. Set up the system of equations. Do not solve.An airline is considering t he purchase of eleven aircraft to meet an estimated demandfor 3200 seats. They have $1540 million to spend on three types of planes: Boeing 747s,Boeing 777s and Airbus A321 s. The table below shows numb er of seats and cost(inmillions) of each plane. How many of each plane should the airline order to meet itsdemand for seats?Seats CostBoeing 747 400 200Boeing 777 300 160Airbus A32 1 200 60WIR Math 166-copyright Joe Kahlig, 10A Page 23. Set up the system of equations. Do not solve.Bob has $82,000 tha t he wants to invest in the stock market and in bonds. Due to thevolatility of the stock market, he has divided t he stocks into two categories: low-riskstocks and high-risk stocks. Bob has decided that the amount invested in high-riskstocks should equal the amount invested in low-risk stocks and bonds combined. Bobestimates that t he bonds will have a dividend rate of 4%/year; low-risk stocks will havea rate of return of 8%/year; and high- risk stocks will have a rate of return of 15%/year.How much money should Bo b invested in each category, if he wa nts to have a returnof $9050/year on the total investment?4. Give the solutions to the system of equations represented by the augmented matrices.(a)x y z1 0 0 210 1 0 90 0 04(b)x y z2 0 0 180 1 0 100 0 530(c)x y z1 0 4 20 1 590 0 0 0(d)x y z w1 2 0 2 70 0 1 4 3(e)x y z1 0 0 40 1 0 20 0 1 80 0 0 0WIR Math 166-copyright Joe Kahlig, 10A Page 35. Perform the row operations that get a zero in the row 2 column 1 po sition and the row3 column 1 position.1 0 912−5 2 1 34 2 −386. Perform the indicated row operations and give the resulting matrix.1 2 537 11 39252 4 5 40 5 6 1(−11)R1+ (2)R2→ R15R3+ (−4)R4→ R37. Solve these systems of equations by any method.(a) 3x + y = 9x + z = 4 + yz − 11 = −3x4x + 6y + 2z = 15 + 7y(b) x + 3y + z = 102x + 7y − z = 214x + 1 3 y + z = 41(c) 3x + 2y + 5z = 7x + 4y + z = 134x − 5y + 2z = −95x + 1 0 y + 7z = 32WIR Math 166-copyright Joe Kahlig, 10A Page 48. Bob has budgeted $840 to expand his video library with 60 DVDs. His local videostore prices their DVDs at $10 for an old release, $16 for a semi-new release, and $22for a current release. How many DVDs from each category will Bob be able to buyassuming that he uses all of the budgeted money?(a) Set up a nd solve the word problem. If the solution is infinite, then place restric-tions on the parameter.(b) How many different solutions are there for this word problem?WIR Math 166-copyright Joe Kahlig, 10A Page 5Section 5.1: Introduction to Matrices• dimension(size) row x columns• aijis the entry in row i and column j• scalar product of a matrix• transpose of a matrix, denoted AT• addition/subtraction o f matrices• must be the same dimension.• if the matrix has a variable in it, then you must do the computation by hand.A ="7 2 46 5 0#B ="9 3 0−1 2 8#C =1 3−2 52 0D ="x 12 5#9. Compute the following with the matrices listed a bove.(a) 3d2,2+ 2c2,1=(b) 3A =(c) CT=(d) A + 2 B =(e) 3A − 4B =(f) 6A − 2B + 3C =(g) 5C − 2 AT=WIR Math 166-copyright Joe Kahlig, 10A Page 610. Solve for x and y.4"6 2xy 5#−"5 3y18 10#="19 x−28
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