SOLUTION: 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, th

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Question 985240: 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?
Found 2 solutions by ikleyn, Edwin McCravy:
Answer by ikleyn(52806)   (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).
Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!
A system of equations is consistent and independent if and only if 
the matrix of coefficients is non-singular, which is the same as saying
the determinant of the matrix of coefficients is not 0.  The two systems
have the same coeficient matrices.  Thus the answer is yes. 


The solutions to the first system 

ax+by=c
dx+ey=f

is




And since it is a consistent and independent system, the denominator ae-bd
is not 0.

The solutions to the second system

ax+by=g
dx+ey=h

 is




the denominator ae-bd is not 0.

So the answer is yes.

Edwin
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