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15 February 2008
Mr. Popik met the team in Mrs. Reilly's room. They played a game called "Stump the Chump"
In "Stump the Chump" student try to present a problem that the "Chump" can't solve. The problems are algebraic equations subject to some constraints. The rules are: no exponents and no hidden exponents. But parentheses, addition, subtraction, multiplication, division, substitution, and variables are allowed. Today, the "Chump" only accepted systems of linear equations. For example, 3 equations with 3 unknowns are allowed. The "Chump" was able to solve all the systems except 2.
There are several ways to solve systems of linear equations: graphing and finding intersection points, substitution, and elimination. There is a nice tutorial explaining these here. There are three possible outcomes with systems of linear equations
- One Solution
- No Solution
- Infinite number of solutions
How can we "Stump the Chump" with a system of equations?
One student managed to have a hidden exponent, so breaking the rules and having a quadratic equation posed a problem for the types of solutions being presented. Using the quadratic formula did result in a solution.
Another way to "Stump the Chump" is to give a set of equivalent equations. If two expressions, which look different, but can be reduced to the same equation are given, then we don't have two equations and two unknowns. We have one equation and one unknown and we can only know a family of solutions.
Example:
(x+4) + (x+3) + (x+2) + (x+1) + x = (y+3)
5x + 8 = y
Can you find the values of x and y? When two equations are redundant, we say that the system is underdetermined. In that case, there is no unique solution.
You can also "Stump the Chump" by having mutually exclusive results. For example, this set of equations:
5x + 8 = y 5x + 3 = y
Here, the equations are parallel lines with different offsets. There are no intersections, and therefore no solutions satisfying both equations.
Yet another way to "Stump the Chump" is to present an overdetermined system of equations. In general, if the number of equations is greater than the number of unknowns, then the system is overdetermined and there is no solution. In such a case, two equations can result in mutually exclusive results.
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