An equation of the form ax + by = c in which a, b and c are known numbers is called a first-degree equation in x and y, or a linear equation in x and y. Thus (A) and (B) are an example of a pair of simultaneous linear equations. The method illustrated above for solving such a pair is called elimination. More specifically, we eliminated y to obtain equation (C).

Eliminating y (or x) from the simultaneous equations

(D)                                               6x - 15y = -10
(E)                                                4x - 10y = -7

leads, upon multiplying both sides of (D) by 2 and both sides of (E) by -3, to

(D')                                             12x - 30y = -20
(E')                                            -12x + 30y = 21

and, upon adding corresponding sides of (D') and (E'), to

(F)                                                          0 = 1

Assuming that there is a pair x, y satisfying (D) and (E) simultaneously has led us to the false conclusion that 0 = 1. Hence there is no pair x, y which satisfies (D) and (E) simultaneously.

9.1 DETERMINANTS OF ORDER 2

Let us now examine the general pair of simultaneous first-degree equations:

We are going to apply the elimination technique discussed above to (G) and then introduce the related concept of a determinant, which has important applications in the theory of systems of equations and in other fields.
 
 

Tuesday, June 23, 1998