98 Algebra. 



Consequently there are hvo sets of values which will satisfy 

 system (a), namely, 



and x-=i7,j= — 12. 



Now the method here used may be applied to the solution of 

 any linear-quadratic system containing two unknowns. In fact, 



take the general case* 



x^ay=b \ ,^ . 



where a, b, c, d, e, f\ and o- are supposed to stand lor any real 

 quantities whatever. 



The value of x in terms of y from the first equation of the 



system is 



x=b—ay (']) 



Substituting this for x in the second equation of the system, that 

 equation becomes 



b^—2aby-\-ay^-^cy'-' + bdy—ady--)reb—aey-{-/y—g. 

 Combining together those terms which contain y^ and those 

 which contain y and transposing all the known terms to the right 

 hand side of the equation, this becomes 



(d' + c—ad)y''-\-(bd—2ab—ae-\-f)y=g—b'—eb, 

 which is a quadratic in which y is the only unknown quantity, 

 whence it can be solved. The values which may be found from 

 this can be substituted in equation (-]) above, and the values of 

 X will be determined. 



12. Examples. Solve the following systems : 



^ f -v+j/=7 



\2x-{-xy-{-2y=i6, 

 f T-^ — 5Xj'H-2J=io 



'^' ] xy=g6. 



•It might be thought that this is not a general case, since x in the first equation and 

 xHn the second do not appear with coefficients. But if either of them had a coefficient 

 the equation could be reduced to the given form by dividing through by that very 

 coefficient. 



