SOLUTION: Solve the system of equations. (If the system is dependent, enter a general solution in terms of c. If there is no solution, enter NO SOLUTION.) 2x − y + z = 12 2y

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: Solve the system of equations. (If the system is dependent, enter a general solution in terms of c. If there is no solution, enter NO SOLUTION.) 2x − y + z = 12 2y       Log On


   

Question 1060073: Solve the system of equations. (If the system is dependent, enter a general solution in terms of c. If there is no solution, enter NO SOLUTION.)
2x − y + z = 12
2y − 3z = −16
3y + 2z = 2
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
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Solve the system of equations.
(If the system is dependent, enter a general solution in terms of c.
If there is no solution, enter NO SOLUTION.)
2x − y + z = 12
2y − 3z = −16
3y + 2z = 2
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

2x -  y +  z =  12,    (1)
     2y - 3z = -16,    (2)
     3y + 2z =   2.    (3)

Notice that the equations (2) and (3) constitute a closed 2x2-sub-system.
   (I use the term "closed sub-system" to highlight that these two equations are for two unknowns y and z and do not include "x".)

For it, multiply eq.(2) by 2 and eq(3) by 3, then add. You will get

4y + 9y + (-6z + 6z) = -32 + 6,   or

13y = -26  --->  y = -2.

Then from (3)  2z = 2 - 3y = 2 - 3*(-2) = 2 + 6 = 8  --->  z = 4.

Next, substitute the found values y and z into equation (1) and get x = 3.


Answer.  x= 3, y= -2, z= 4.