In certain mathematician circles, **C!** is also a shorthand term for "contradiction." The clearest example of its proper usage can be found in a problem involving solving a system of equations.

Suppose that you are given the following system to solve:

(i) x + 4y - z = 4

(ii) 3x - 2y + 5z = 6

(iii) 5x + 6y + 3z = 13

One solution^{1} to this system involves eliminating one of the variables and then adding or subtracting the resulting equations. Let us add (iii) with 3*(i).

3*(i) 3x + 12y - 3z = 12

(iii) __5x + 6y + 3z = 13__

(iv) 8x + 18y = 25

We would then add 5*(i) to (ii) to garner a second equation without a "z" term.

5*(i) 5x + 20y - 5z = 20

(ii) __3x - 2y + 5z = 6__

(v) 8x + 18y = 26

Subtracting (iv) from (v) produces the result 0=1 which immediately implies a contradiction. Namely **C!**. Therefore, a solution with only contradictory solutions **C!** has no solutions.

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^{1}Other solution methods include but are not limited to the iterative method, back substitution, matrices, and graphs.