Dr. Sarah's Maple Commands and Polya's How to Solve it for Problem Set 1

I will post on ASULearn answers to select questions I receive via messaging or in office hours. Be sure to carefully follow the guidelines and both the online and book directions in order to receive full credit. Also see the Maple Tips. Print Maple and/or show by-hand work, and annotate with your reasoning. Reviewing class notes and Exercise Solutions that are on ASULearn will often help you with problem sets.

Execute at the Start of Every 2240 Maple

> with(plots): with(LinearAlgebra):

Plotting a System of Equations

> implicitplot({2*x+4*y-2,5*x-3*y-1}, x=-1..1, y=-1..1);

2 lines that intersect in one point.

> implicitplot3d({x+2*y+3*z-3,2*x-y-4*z-1,x+y+z-2},x=-4..4,y=-4..4,z=-4..4);

3 planes that intersect in one line (infinite solutions).

Augmented Matrix and Gauss-Jordan (or Reduced Row Echelon) Form

> A:=Matrix([[-1,2,1,-1],[2,4,-7,-8],[4,7,-3,3]]); ReducedRowEchelonForm(A);

Hence we see that the solutions are a point that the 3 planes intersect in: (195/83, -15/83, 142/83).

Gaussian (or Row Echelon) Form

Note that if there are unknown variables in the matrix, we should use GaussianElimination instead.

> P:=Matrix([[1,3,4,k],[2,8,9,0],[10,10,10,5],[5,5,5,5]]); GaussianElimination(P);

Analysis will show that this is impossible - the 4th row tells us 0x+0y+0z=5/2, which is impossible. The 4 planes don't have a concurrent intersection.

Notice that there are three different systems with 3 variables above. The first (implicitplot3d) has infinite solutions, the second (ReducedRowEchelonForm) has 1 unique solution, and this last one has 0 solutions.

Mathematics, you see, is not a spectator sport. [George Polya, How to Solve it]

Understand the problem:
Do you understand all the words used in the problem? Review them in ASULearn solutions, the ASULearn glossary, and the book, or ask me!
What are you asked to find or show?
Can you restate the problem in your own words?

Devise a plan and carry it out:
Polya mentions that there are many reasonable ways to solve problems. The skill at devising and carrying out an appropriate strategy is best learned by practicing and solving problems yourself. Strategies include
Look for a pattern
Draw a picture
Use a model
Consider special cases
Use a formula
Solve an equation
Guess and check
Be ingenious

Look back:
Much can be gained by taking the time to reflect and look back at what you have done, what worked, and what didn't. Doing this will enable you to predict what strategy to use to solve future problems. Also look back to ensure that you have answered all parts of the question. Finally think about connections or extensions.