Presentation on theme: "Applications of Linear Systems"— Presentation transcript:
1Applications of Linear Systems 1.2 Day 2Applications of Linear Systems
2Is the following matrix in reduced row echelon form? How many solutions are there to the system of equations representedThis matrix?
3Solution The matrix represents the following equations. Here are equations that represent the solutions to they systemIn matrix language they are expressed as:orWe can find particular values by plugging in arbitrary values for s and t
4Problem Pre-CalcFill in the blank to make the two matrices row equivalent
5Problem 20 (1.1)Consider an economy that has 2 industries A and B. Assume that the consumer demand for their products is 1,000 and 780 respectively (in millions of dollars). What outputs should the two industries generate to satisfy the demand?(see next slide)
6Problem 20 ContinuedYou might be tempted to say 1,000 and 780 respectively. However, we must consider the industry demand as well. If industry A produces say electricity then industry B needs 10 cents ($.1) worth of electricity for every $1 of output. Similarly Industry A needs $.2 worth of B’s products for every $1 of out put. The out put of A and the output of B must satisfy both the industry demand and the consumer demand.
7Problem 20 solutionThe total demand for the product of Industry A is the consumerdemand 1000 plus .1b The demand from industry b. The out putmust meet this demand.Setting up s similar equation for b we get the systema= bb= aOra - .1b = 1000-.2a + b = 780Which yields the solution a = 1100 and b = 1000We will revisit this problem later with more complex interactions
8Problem 72 Pre-Calc 8.1Find the values of a,b, and c so that the parabola goes through the given points
9Solution to problem 72 pre-Calc Substitute in x and y to obtain the following equationsa(1) + b(1) + c = 9a(4) + b(2) + c = 8a(9) + b(3) + c = 5This can be thought of as the following matrixAnd solved on a TI 89 graphing calculatorrref([1,1,1,9;4,2,1,8;9,3,1,5])
108.1 Pre-Calc bookWrite a systems of equations so that the Quartic (4th degree equation passes through the given points) Use technology to solve they system(solution similar to last problem) not presented here)
11Pre- Calc ExampleOne major area of applications for matrices is for networks. Write a matrix that represents network use technology to solve the system. Interpret the meaning of the answer
12Solution to Example10Let x5 = t, where t is a real number, you have x1 = t – 10,x2 = - t + 30, x3 = t -10, x4 = t + 10, so this system has an infinitenumber of solutions