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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
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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.