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

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)   (Show Source): You can put this solution on YOUR website!

If the system

          (1)

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

          (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 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).

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