If we multiply both sides of the first equation in (G) by b₂ and both sides of the second equation by b₁, we obtain

Whence, by subtraction, we obtain:

(1)                                                        (a₁b₂ - a₂b₁)x = c₁b₂ - c₂b₁.

If we use the equations in (G) to eliminate x rather than y, the result is

(2)                                                       (a₁b₂ - a₂b₁)y = a₁c₂ - a₂c₁.

If a₁b₂ - a₂b₁ is not zero, we see that the solution of the system (G) is found from (1) and (2) in the form

We note that the denominators are the same, that the numerator in the expression for x is obtained from the denominator by replacing the coefficients a₁ and a₂ of x in (G) with c₁ and c₂, respectively, and that the numerator in the expression for y is obtained from the denominator by replacing the coefficients b₁ and b₂ of y with c₁ and c₂.

This motivates us to introduce the notation

The square array bordered by vertical lines in (4) is called a two-by-two (2x2) determinant. With this notation, the equations in (3) can be rewritten as

provided that the common denominator is not zero.
 
 

Tuesday, June 23, 1998