Find the Solution

Click here to open a new window with a live diagram that you can use to explore the system of linear equations

or, in matrix form,

The blue ball represents a particular value of the vector x that is a possible solution of this system of equations. If this particular value is a solution then the vector y = Ax represented by the red ball will be right on top of the black square representing the vector b that represents the two constants appearing on the right side of this system of equations. You can try different possible solutions by dragging the vector x around until the vector y is right on top of the vector b. It may be difficult to do this exactly but you should be able to get close.

Questions:

  1. Find the solution to the system of equations above as described above. Then solve it either algebraically or by using technology. Compare your answers. Note that because the action of the live figure is limited by the resolution of the screen you may not be able to get the exact answer by dragging the vector x.

  2. You can edit the entries in the live figure for the constants b1 and b2 in the usual way. Change these entries to study the system of linear equations:

    Does this system have a solution? If so, does it have only one solution or does it have an infinite number of solutions?

  3. You can edit the entries in the live figure for the matrix A in the usual way. Change these entries and the entries for the constants b1 and b2 to study the system of linear equations:

    Does this system have a solution? If so, does it have only one solution or does it have an infinite number of solutions?

  4. Change the entries in the live figure to study the system of equations

    Does this system have a solution? If so, does it have only one solution or does it have an infinite number of solutions?

  5. Change the entries in the live figure to study the system of equations

    Does this system have a solution? If so, does it have only one solution or does it have an infinite number of solutions?

  6. Do you have any thoughts or observations about when systems of equations have no, one, or infinitely many solutions?