Let us consider a pair of simultaneous equations:

(A)                                                  2x - 3y = 9
(B)                                                  8x + 5y = 7
 

We seek numbers x and y which satisfy (A) and (B) simultaneously, that is, a pair x,y for which both (A) and (B) become true statements. Such a pair also satisfies the equations

(A')                                              10x - 15y = 45
(B')                                              24x + 15y = 21

obtained by respectively multiplying both sides of (A) by 5 and both sides of (B) by 3.  A pair x, y satisfying both (A') and (B') has to satisfy

(C)                                                     34x = 66

which is obtained by adding corresponding sides of (A') and (B'). The only root of (C) is
x = 66/34 = 33/17.

If one replaces x by 33/17 in (A), one finds that

Hence the only pair of numbers x,y which might satisfy (A) and (B) simultaneously is
x = 33/17, y = -29/17. Our method of obtaining these numbers shows that they do satisfy (A). We check that they also satisfy (B) by substituting in the left side as follows:

This shows that x = 33/17, y = -29/17 is the unique pair that satisfies (A) and (B) simultaneously.
 
 

Wednesday, June 10, 1998