P^{*}_{1} - P^{*}_{1}(T) = 0 |
P y_{1} - x_{1} P^{*}_{1} = 0 |
P^{*}_{2} - P^{*}_{2}(T) = 0 |
P y_{2} - x_{2} P^{*}_{2} = 0 |
x_{1} + x_{2} - 1 = 0 |
y_{1}+ y_{2} - 1 = 0 |
P^{*}_{1} - P^{*}_{1}(T) = 0 | (1) |
p_{1} - x_{1} P^{*}_{1} = 0 | (2) |
P^{*}_{2} - P^{*}_{2}(T) = 0 | (3) |
p_{2} - x_{2} P^{*}_{2} = 0 | (4) |
x_{1} + x_{2} - 1 = 0 | (5) |
p_{1}+ p_{2} - P = 0 | (6) |
The remaining unknowns in the equations are then:
P , P^{*}_{1} , p_{1} , P^{*}_{2} , x_{2} , p_{2}
We can draw up an incidence or occurrence table for the set of equations as shown below, with a row for each equation and a column for each unknown, where an `x' indicates that the specified variable occurs in the equation and a blank that it does not.
Variable: | P | P^{*}_{1} | p_{1} | P^{*}_{2} | x_{2} | p_{2} |
Equation (1) | x | |||||
Equation (2) | x | x | ||||
Equation (3) | x | |||||
Equation (4) | x | x | x | |||
Equation (5) | x | |||||
Equation (6) | x | x | x |
We might want to write this in `mathematical' notation we would replace the x's by ones, leaving the blanks, which are conventionally taken to imply zeros. The matrix would then look like this:
We explain the use of this type of matrix or table for ordering equations in section 1.1.2.3. To summarise briefly, the matrix shows that equations (1), (3) and (5) contain only a single unknown each, and so can be solved for each of these, i.e. respectively for P^{*}_{1} , P^{*}_{2} and x_{2}. It can then be seen that equation (2) now contains only one remaining unknown, p_{1} as P^{*}_{1} is no longer unknown, and so it can be solved. In fact all the remaining unknowns can now be determined by solving equation (4) and then (6) each for a single unknown.
Use of the matrix has shown how a problem which might have been thought to involve the simultaneous solution of 6 equations in six unknowns actually involves solving only one equation at a time.