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Exercise 1:
After you've toiled over solving systems of
three equations and three unknowns by hand, we'll bet you're ready for some instant
answers! Go check out this Linear Equations
Solver that will solve these systems in the blink of an eye. Each equation is to be
entered in the form:ax + by + cz = d
The blanks are set up so that you
enter the values for a, b, c and d. Your mission is to use this site to help you find two
systems that are dependent and two systems that have no solution. (For the latter, the
site will say "NULL" -- this is just another way of saying that the solution set
is empty.) |
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Exercise 2:
Finding solutions using Gauss-Jordan
Elimination is a lot of hard work, isn't it? (Even though it *is* a lot of fun!) This
process is really intended for solving really large systems of equations with lots of
unknowns via computers. Would you want to solve this by hand?
0.7v + 5/8w - 78x +
345.7y - z = 23
1/2v - 89w + 89/7x + 42.375y + 10z = 7
3v + 7w - 4x + 18y = 0
45v - 4.32w + 8/77x - 22y + 5.4z = -2
9v + 9w + 9x + 9y + 9z = 8 |
Go to The Java Matrix Solver and solve
this system. Be sure to read through their example and instructions. Just put a blank
space between the entries of each row and hit ENTER (on your keyboard) between the rows.
The site only gives decimal solutions... But, if we were figuring out how much rocket fuel
to use, 3.14159 gallons would be just as good as pi gallons. |
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Exercise 3:
In section 8-3, you learned about solving
systems of nonlinear equations. We can see that our solutions are simply the points of
intersection between the graphs. The Quadratic Curve Applet
will allow you to play with the solutions (intersections) of various lines and parabolas.
(In calculus, you will be asked to find the area between these curves and you'll need to
be able to find the intersections to accomplish this goal.) You can move the individual
graphs by dragging the squares and you can change their shapes by dragging the circles. Be
sure that your browser window is maximized so that you can see the information to the
right of the graph. Keep a record of your work by writing down your equations and drawing
the graphs by hand.
| 1) |
Draw a graph that has two distinct
solutions. |
| 2) |
Draw a graph that has only one
solution. |
| 3) |
Draw a graph that has no
solutions. |
| 4) |
Draw a graph that has two distinct
solutions with the y coordinates being the same in both. What type of line did you have to
draw? |
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