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

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.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.