SOLUTION: Please help me with this true or false question: True or False: The system: ax+by=c dx+ey=f is a consistent system of independe

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Question 985235: Please help me with this true or false question:
True or False:
The system:
ax+by=c
dx+ey=f
is a consistent system of independent equations (with variables x and y). When c and f are replaced by two new numbers g and h, the new system of equations is:
ax+by=g
dx+ey=h
Is the new system also a consistent system of independent equations?
Thank you in advanced.
Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!

If the system

system%28ax%2Bby=c%2C%0D%0Adx%2Bey=f%29           (1)

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

system%28ax%2Bby=g%2C%0D%0Adx%2Bey=h%29           (2)

is a consistent system of independent equations too.

Algebraic explanation
The system  (1)  is a consistent system of independent equations if and only if its determinant  det %28matrix%282%2C2%2C+a%2C+b%2C+d%2C+e%29%29  is non-zero.
Therefore,  if the system  (1)  is a consistent system of independent equations,  then the system  (2)  is too,  because both the systems have the same matrix.

Geometric explanation
The system  (1)  is a consistent system of independent equations if and only if two straight lines  ax + by = 0  and  dx + ey = 0 are distinct and non-parallel
(then they intersect in some unique point).

Therefore,  if the system  (1)  is a consistent system of independent equations,  then the system  (2)  is too,  because the system  (2)  represents
the same straight lines as the system  (1).