Visualizing Systems of Linear Equations This worksheet is to accompany Section 1.1, Lay's Linear Algebra Based on a worksheet by Michael K. May, revised by Russell Blyth. Revised by Harry S. Mills. restart: with(LinearAlgebra): with(plots): with(plottools):

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Outline: The basic objectives are: 1) Use Maple to visualize the solution to a system of linear equations in R2. LinearAlgebra, plots, and plottools packages solve command used to solve system and to solve for a single variable for plotting plot command 2) Use Maple to visualize the solution to a system of linear equations in R3. display command seq command showplanes procedure

Consider the system of equations: eq1 := x + y = 3; eq2 := x - y = -1;

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It is a simple exercise from college algebra to solve the system, obtaining the solution {x=1, y=2}. However, in this class, we are not as interested in solving this simple problem as we are in setting up a framework of understanding that we can use to attack larger systems of linear equations. Nevertheless, it is worthwhile to note that Maple will solve the system with the solve command. solve({eq1,eq2});

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The same solve command lets us solve a single equation for a specified variable. In particular, we can solve eq1 for y. We can get the answer either as an expression, or in equation form. solve(eq1,y);solve(eq1,{y});

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We can also plot the system and see its solution. plot({solve(eq1,y),solve(eq2,y)},x=-5..5,y=-5..5);

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

It should be noted that if we add in extra equations that are linear combinations of the initial set, then we will get the same solution. plot({solve(eq1,y),solve(eq2,y),solve(eq1-eq2,y), solve(2*eq1+eq2,y)}, x=-5..5,y=-5..5);

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A system has one solution when all of the lines meet in a single point. To have more then one solution, all of the lines must meet together in at least two points. Since two points define a line, that would mean that the equations all define the same line. If we have only two lines in the system, the only way for there to be no solution is for the lines to be parallel. eq3 := x + y = 3; eq4 := 2*x + 2*y = 6; solve({eq3,eq4}); plot({solve(eq3,y),solve(eq4,y)},x=-5..5,y=-5..5);

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

The two equations above describe the same line, and so the solution set for the two equations is that whole line. eq5 := x + y = 3; eq6 := x + y = 1; solve({eq5,eq6}); plot({solve(eq5,y),solve(eq6,y)},x=-5..5,y=-5..5);

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Since the two lines have no common point the solution set is empty. Note that Maple gives an empty response to the solve command. Three or more lines can meet pairwise in distinct points. Consider the following system: eq7 := x + y = 4; eq8 := x - y = 2; eq9 := 2*x - y = 1; plot({solve(eq7,y), solve(eq8,y), solve(eq9,y)}, x=-5..5, y=-5..5);

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Any two lines meet, but the whole system has no solution.

1) Solve the system {2x-y=4, x+2y=-4} three ways, first with pen and paper using college algebra methods, then with the solve command using Maple, then graphically using Maple. 2) Attempt to solve the system {x+y=2, -x+y =2, 2x+y=-2} the same three ways.

If we try the same approach to visualizing systems of linear equations in three unknowns, we run into a minor display problem. eq10 := 2*x + 3*y + z = 2; eq11 := 2*x + 4*y + 7*z =6; eq12 := 3*x + 10*y + 5*z = 7; solve({eq10,eq11,eq12});

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plot3d({solve(eq10,z),solve(eq11,z),solve(eq12,z)}, x=-5..5,y=-5..5, view=-5..5, axes=boxed);

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When Maple graphs 3D objects, by default it uses a color scheme that is the same for all objects. Maple does this to make it easy to perceive depth in the 2D display of a 3D object. Unfortunately, it makes it difficult to see intersections. We have to create plots in different colors for each of the equations, and then display them together. display({ plot3d(solve(eq10,z),x=-5..5, y=-5..5, color=red), plot3d(solve(eq11,z),x=-5..5, y=-5..5, color=blue), plot3d(solve(eq12,z),x=-5..5, y=-5..5, color=green)}, view=-5..5, axes=boxed);

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