Introduction to Systems of Equations
In this chapter, we illustrate the use of systems of equations to solve real- world applications from the sciences, engineering, and economics. Previously, we have seen examples of linear systems in solving discrete dynamical systems and in model fitting with least squares. Maple’s LinearAlgebra package has a number of functions and programs that will be very useful for solving these problems.
There are exactly three possibilities for a system of linear equations: the system has a unique solution, an infinite number of solutions, or no solution. Consider the following two-dimensional system of linear equations:
The augmented matrix for this system is
Row reduce the augmented matrix to find one of the three possible results. Unique Solution. Row reduction of A gives
which yields the unique solution (x,y) = (d, e).
Infinitely Many Solutions. Row reduction of A gives one of three possibilities. First,
which yields solutions (x,y) = (a ,c — b ) with arbitrary t. Second, if ai = 0 and b| ф 0, then row reduction gives
and the solution is (t, d) for arbitrary t. Third, if «1 ф 0 and bi = 0, then row reduction gives
and the solution is (d. t) for arbitrary t.
No Solution. Row reduction gives
with e^O. Then there is no solution as Ox + Oy = e ф 0 is impossible. These alternatives are represented visually in Figure 6.4.
FIGURE 5.4: The Three Alternatives for Solutions of a Linear System
Describe the possibilities for three linear equations in three dimensions; that is, for the intersection of three planes in 3D.
We will rely on the matrix forms in recognizing our solutions in this chapter.
The following description is from the Details of the LinearAlgebra Package Help page.[1]
- • Maple’s Linear Algebra package is an efficient and robust suite of routines for doing computational linear algebra.
- • Full rectangular and sparse matrices are fully supported at the data structure level,[2] [3] as well as upper and lower triangular matrices, unit triangular matrices, banded matrices, and a variety of others. Further, symmetric, skew-symmetric, hermitian. and skew-hermitian are known qualifiers that are used appropriately to reduce storage and select amongst algorithms.
According to V. Z. Aladjev,
The LinearAlgebra module has been designed to accommodate different sets of usage scenarios: casual use and programming use. Correspondingly, there are functions and notations designed for easy casual use (sometimes at the cost of some efficiency), and some functions designed for maximal efficiency (sometimes at the cost of ease- of-use). In this way, the LinearAlgebra facilities scale easily from first-year classroom use to heavy industrial usage, emphasizing the different qualities that each type of use needs.0
The LinearAlgebra commands we will use most are:
GaussianElimination: performs Gaussian elimination on the matrix A and returns an upper triangular matrix U.
ReducedRowEchelonForm: performs Gaussian elimination on the matrix A and returns the reduced row echelon form of A.
(Note for those who have studied linear algebra: Both of these commands use LinearAlgebra's LUDecomposition function to determine the upper triangular factor of A.)
Recall there are many ways to enter a matrix. One of the easiest is to use Maple’s Matrix palette. Enter help(’worksheet, matpalette’) for details.
Begin with a simple 3x3 system of linear equations.
Example 5.1. A 3 x 3 System.
Determine the solution of the following system of linear equations.
Load the linear algebra package, enter the system’s matrix in augmented form, row reduce the matrix using Gaussian elimination, and then interpret the results.
We interpret the results as a unique solution: (x,y,z) = (2,1,-1).
Let’s increase the dimension by one.
Example 5.2. A 4 x 4 System.
Determine the solution of the following system of linear equations.
We interpret the results: The system has no solution as the last row of the row reduced augmented matrix gives the equation 0=1.
Exercises
Use Maple to solve the following system of equations and interpret the results.
- 1. x-by=-lbA x — 3y = —84
- 2. 1 la; — 6y = 494
x + 7у = —23
- 3. 9x + у = 56 6x - by = 128
- 4. 6x + у = 50 18a; + 3y = 150
- 5. x + 2 у + Зл = 5 x — у + 6z = 1
Зж — 2 у =4
- 6. х + у + 3z — Aw = 12 Зх -- у — 2z — w= 0
- 7. 2х + Зу + Az = 5
х — у + 2z = 6 Зх — by — z = 0
8. 2х — Зу + 2z = 21
х + Ay — z = 1 —х + 2 у + z = 17
9. Зж — Ay + bz — Aw = 12
ж — у + z — 2 w = 0 2ж + у + 2z + 3 w = 52 2ж — 2y + 2z — 3 w = 1
Projects
Project 5.1. Model a solution methodology for Ax = b using linear algebra and illustrate your method with Maple commands.
Project 5.2. Let A be a random n x (n + 1) matrix representing a linear system of n linear equations in n variables. Estimate the probability that the system has a unique solution when
- (a) n = 3.
- (b) n = 4.
- (c) n = 5.
- (d) n > 5.
