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

Algebra ->  Matrices-and-determiminant -> 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       Log On


   

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

ax + by = c
dx + ey = f

By Cramer's rule:

Delta = abs%28matrix%282%2C2%2Ca%2Cb%2Cd%2Ce%29%29 = ae - bd

Dx = abs%28matrix%282%2C2%2Cc%2Cb%2Cf%2Ce%29%29 = ce - bf 

Dy = abs%28matrix%282%2C2%2Ca%2Cc%2Cd%2Cf%29%29 = af - cd

x = D%5Bx%5D%2F%28Delta%29 = abs%28matrix%282%2C2%2Cc%2Cb%2Cf%2Ce%29%29%2F%28ae-bd%29 = %28ce-bf%29%2F%28ae-bd%29

y = D%5By%5D%2F%28Delta%29 = abs%28matrix%282%2C2%2Ca%2Cc%2Cd%2Cf%29%29%2F%28ae-bd%29 = %28af-cd%29%2F%28ae-bd%29

Denominators cannot be 0, so Delta = abs%28matrix%282%2C2%2Ca%2Cb%2Cd%2Ce%29%29 = ae - bd ≠ 0.
and both %28ce-fe%29%2F%28ae-bd%29 and %28af-cd%29%2F%28ae-bd%29 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 = %28ce-bf%29%2F%28ae-bd%29

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 = %28af-cd%29%2F%28ae-bd%29

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