Question 985235


If the system 


{{{system(ax+by=c,
dx+ey=f)}}}           (1)


is a consistent system of independent equations  (with variables  x  and  y),  then the system 


{{{system(ax+by=g,
dx+ey=h)}}}           (2)


is a consistent system of independent equations too.


<B><U>Algebraic explanation</U></B><BLOCKQUOTE>The system &nbsp;(1)&nbsp; is a consistent system of independent equations if and only if its determinant &nbsp;det {{{(matrix(2,2, a, b, d, e))}}}&nbsp; is non-zero.

Therefore, &nbsp;if the system &nbsp;(1)&nbsp; is a consistent system of independent equations, &nbsp;then the system &nbsp;(2)&nbsp; is too, &nbsp;because both the systems have the same matrix.</BLOCKQUOTE> 

<B><U>Geometric explanation</U></B><BLOCKQUOTE>The system &nbsp;(1)&nbsp; is a consistent system of independent equations if and only if two straight lines &nbsp;ax + by = 0 &nbsp;and &nbsp;dx + ey = 0 are distinct and non-parallel 

(then they intersect in some unique point).


Therefore, &nbsp;if the system &nbsp;(1)&nbsp; is a consistent system of independent equations, &nbsp;then the system &nbsp;(2)&nbsp; is too, &nbsp;because the system &nbsp;(2)&nbsp; represents 

the same straight lines as the system &nbsp;(1).</BLOCKQUOTE>