Reaction balancing requires a characteristic system of equations, but its coefficient matrix should be of rank . In fact, the coefficients are not fixed unambiguously but are proportional to each other. In this case, we choose the traditional convention of using the lowest possible positive integer values. Moreover, in order for the results to be physically valid, the coefficients in the chemical equation must be positive.

For example, the reaction

has ; for , , and for , (2 equations in 3 variables). The coefficient matrix is

.

By convention, the coefficients for the reactants are taken to be positive and for the products, negative; the matrix rank is equal to 2. The simplex is constructed from the columns of the coefficient matrix. In dimensions, the simplex has as vertices the columns of subscripts in the chemical reaction formula.

For the reaction to occur, the interior of the simplex must include the origin [1].

The reactions considered here involve two species (such as and in , , …), which generate a segment (a 1-simplex), or three species (such as , , in , ), which generate a triangle (2-simplex). As a result, in all possible cases, a balanced reaction is obtained.

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