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Systems of Linear Equations

Part 5: Geometric Interpretations of the Algebraic Equations

  1. We consider first the system

      x1 + 2x2 = 16
      x1 - x2 = 2

    The worksheet is set up both to solve this system and to graph the lines that represent each of the equations. Which point in the graph corresponds to the solution of the system?

  2. Next we generalize our system to any system in two variables with the same constant terms:

      ax1 + bx2 = 16
      cx1 + dx2 = 2

    The computer algebra system will solve this system for you whether the coefficients have values or not. When you set the values of the coefficients as they were originally and re-enter the solve command, you should get the same solution as before. Also, a generalized version of the plot command will draw the same picture as before. The point of the generalized commands is that you can change the values of the coefficients and see how the numerical solution and the graph change. Experiment with several changes in the coefficients to see what happens.

  3. To see an inconsistent system, change the value of a to -2 and leave (or set) the other coefficients as they were to start.

  4. To see a system with an infinite number of solutions, change the values of a to 8 and b to -8.

  5. We turn next to a geometric interpretation of systems with three unknowns. Recall that any linear equation of the form

      ax1 + bx2 + cx3 = d

    has a graph that is a plane perpendicular to the vector (a, b, c). In the worksheet, we show first the graph of

      x1 + 4x2 + 3x3= 10.

    Then we add to the picture the graph of

      2x1 + x2- x3 = -3.

    Finally, we add the graph of

      x1 - x2 + x3 = 3.

    What point in this last picture corresponds to the solution of the system

      x1 + 4x2 + 3x3 = 10,
      2x1 + x2 - x3 = -3,
      x1 - x2 + x3 = 3       ?

  6. Now adapt the 3-D plot command to study the geometry of the following systems studied in Parts 1 and 3. If necessary, vary the ranges for x1, x2 and x3 (view), as well as the orientation, until you are sure you see what's going on.

      2x1 + 8x2 + 6x3 = 20
      4x1 + 2x2 - 2x3 = -2
      3x1 - 2x2 + x3 = 11

      x1 + 4x2 + 3x3 = 10
      2x1 + x2 - x3 = -1
      3x1 - x2 - 4x3 = 11

  7. If a system of three equations in three unknowns has a unique solution, what can we say about the three planes the system represents?

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