SOLUTION: Will someone help me to solve the system of equations by using addition (elimination)method. If the answer is a unique solution, present it as an ordered pair: (x, y). If not, spec

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Question 145910This question is from textbook Intermediate Algebra
: Will someone help me to solve the system of equations by using addition (elimination)method. If the answer is a unique solution, present it as an ordered pair: (x, y). If not, specify whether the answer is "no solution" or "infinitely many solutions.
2x – 7y = 3
-4x + 14y = -9
This question is from textbook Intermediate Algebra

Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!
Will someone help me to solve the system of equations by using addition (elimination)method. If the answer is a unique solution, present it as an ordered pair: (x, y). If not, specify whether the answer is "no solution" or "infinitely many solutions.

 2x –  7y =  3
-4x + 14y = -9 

To make the y's cancel, multiply the top equation 
through by 2:  4x - 14y = 6


 4x - 14y =  6
-4x + 14y = -9

Now add vertically term by term:

 4x - 14y =  6
-4x + 14y = -9
--------------
        0 = -3

All the variables canceled out and left a FALSE 
numerical equation.  Therefore there is no solution,
and the system is called "inconsistent".

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

However if you had had this system of equations instead,
where there were a -6 where the -9 is:

 2x –  7y =  3
-4x + 14y = -6 

To make the y's cancel, you would, as above multiply the 
top equation through by 2:  4x - 14y = 6


 4x - 14y =  6
-4x + 14y = -6

Now as above, you would add vertically term by term:

 4x - 14y =  6
-4x + 14y = -6
--------------
        0 =  0

As above, all the variables canceled out, but this time
we are left with a TRUE numerical equation.  Therefore 
there are infinitely many solutions, and the system
is called "dependent".

Edwin


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