SOLUTION: Tell how to determine whether the system has a unique solution, and solve it with the Cramer method: ax + by = c dx + ey = f The answer is: the system has a unique solution

Question 549768: Tell how to determine whether the system has a unique solution, and solve it with the Cramer method:
ax + by = c
dx + ey = f
The answer is: the system has a unique solution if:
delta = ae - bd not equal to 0
and the solution is:
x = (ce - bf) / (ae - bd)
y = (af - cd) / (ae - bd)
How do we explain that?

Answer by Edwin McCravy(20060) (Show Source): You can put this solution on YOUR website!


ax + by = c
dx + ey = f

By Cramer's rule:

Delta =  = ae - bd

D_x =  = ce - bf 

D_y =  = af - cd

x =  =  = 

y =  =  = 

Denominators cannot be 0, so Delta =  = ae - bd ≠ 0.
and both  and  have unique
values when that denominator is not zero.

------------------------------------------------------

In case you need to explain why Cramer's rule works:
 
To eliminate y, multiply the first equation by e and the
second equation by -b, and add the equations vertically
term by term:

     aex + bey =  ce
    -bdx - bey = -bf
-----------------------
 aex-bdx       =  ce-bf  
(ae-bd)x       =  ce-bf
       x = 

To eliminate x, multiply the first equation by -d and the
second equation by a, and add the equations vertically
term by term:

  -adx   - bdy = -cd
   adx   + aey =  af
-----------------------
       aey-bdy =  af-cd  
      (ae-bd)y =  af-cd
             y = 

a and y are the same using the elimination method as they are 
using Cramer's rule.
  
Edwin

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