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 1 9
4 2 1 8
9 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