APPENDIXE
Solution of Simultaneous Equations
CRAMER’S RULE
Cramer’s rule provides an efficient organization for the work that is needed to solve a set of
simultaneous linear algebraic equations. Here we develop formulae for the cases of two and three
unknowns. When there are more than three unknowns, the arithmetic becomes quite tedious; in
such cases, the solution is best carried out by a computer or calculator program.
Let us consider a pair of linear algebraic equations with two unknowns,x 1 andx 2 , written in
the form
a 11 x 1 +a 12 x 2 =b 1 (1a)
a 21 x 1 +a 22 x 2 =b 2 (1b)
Cramer’s rule gives the solution for the unknowns as
x 1 =D 1 /D (2a)
x 2 =D 2 /D (2b)
where theDs are thedeterminantsgiven byD=∣
∣
∣
∣a 11 a 12
a 21 a 22∣
∣
∣
∣=a^11 a^22 −a^12 a^21 (3a)D 1 =∣
∣
∣
∣b 1 a 12
b 2 a 22∣
∣
∣
∣=b^1 a^22 −a^12 b^2 (3b)andD 2 =∣
∣
∣
∣a 11 b 1
a 21 b 2∣
∣
∣
∣=a^11 b^2 −b^1 a^21 (3c)
The extension of Cramer’s rule to more than two equations is very similar to the results
for two equations, but is slightly more involved in the evaluation of the resulting determinants.
For example, let us consider a set of three linear simultaneous algebraic equations with three
unknowns,x 1 ,x 2 , andx 3 :
a 11 x 1 +a 12 x 2 +a 13 x 3 =b 1 (4a)
a 21 x 1 +a 22 x 2 +a 23 x 3 =b 2 (4b)
a 31 x 1 +a 32 x 2 +a 33 x 3 =b 3 (4c)
Cramer’s rule yields the solution for the three unknowns as
xk=Dk/D , k= 1 , 2 , 3 (5)
where
D=∣ ∣ ∣ ∣ ∣ ∣
a 11 a 12 a 13
a 21 a 22 a 23
a 31 a 32 a 33∣ ∣ ∣ ∣ ∣ ∣=a 11∣
∣
∣
∣a 22 a 23
a 32 a 33∣
∣
∣
∣−a^12∣
∣
∣
∣a 21 a 23
a 31 a 33∣
∣
∣
∣+a^13∣
∣
∣
∣a 21 a 22
a 31 a 32∣
∣
∣
∣=a 11 (a 22 a 33 −a 23 a 32 )−a 12 (a 21 a 33 −a 23 a 31 )+a 13 (a 21 a 32 −a 22 a 31 ) (6a)843