Applications of Linear Systems 1.2 Day 2 Applications of Linear Systems

Is the following matrix in reduced row echelon form? How many solutions are there to the system of equations represented This matrix?

Solution The matrix represents the following equations. Here are equations that represent the solutions to they system In matrix language they are expressed as: or We can find particular values by plugging in arbitrary values for s and t

Problem 20 8.1 Pre-Calc Fill in the blank to make the two matrices row equivalent

Problem 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)

Problem 20 Continued You 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.

Problem 20 solution The total demand for the product of Industry A is the consumer demand 1000 plus .1b The demand from industry b. The out put must meet this demand. Setting up s similar equation for b we get the system a= 1000 + .1b b= 780 + .2 a Or a - .1b = 1000 -.2a + b = 780 Which yields the solution a = 1100 and b = 1000 We will revisit this problem later with more complex interactions

Problem 72 Pre-Calc 8.1 Find the values of a,b, and c so that the parabola goes through the given points

Solution to problem 72 pre-Calc Substitute in x and y to obtain the following equations a(1) + b(1) + c = 9 a(4) + b(2) + c = 8 a(9) + b(3) + c = 5 This can be thought of as the following matrix 1 1 9 2 1 8 3 1 5 And solved on a TI 89 graphing calculator rref([1,1,1,9;4,2,1,8;9,3,1,5])

8.1 Pre-Calc book Write 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)

Pre- Calc Example 10 8.1 One 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

Solution to Example10 Let 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 infinite number of solutions

Problem 30

Problem 30 Solution

Homework p. 6 #21, 23,29,31,37,40,42 p. 20 #37 Pre-Calc p. 562 19,21,79,81,83,85