There are three variables and only two equations. Therefore we can't get a single solution; we can only get a family of solutions in terms of some parameter.
The process is straightforward, although the calculations are generally a bit ugly.
(1) Use elimination to reduce the system of 2 equations and 3 unknowns to a system of 1 equation with 2 unknowns.
(2) Solve that single equation for one variable in terms of the other.
(3) Substitute into either of the original equations to find an expression for the third variable.
(1) I chose to eliminate x: multiply the first equation by 2 and the second equation by -3 and add:
6x - 8y - 14z = -12
-6x - 9y + 15z = -3
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-17y + z = -15
Solve for z in terms of y:
z = 17y - 15
(3) Substitute y for y and 17y-15 for z in one of the original equations to find x in terms of y:
ANSWER:
x = 41y-37; y = y; z = 17y - 15
Choose any number for the "parameter" y and use it to find values for x and z. Those values of x, y, and z will satisfy both of the given equations.