Elimination always bothers me with systems of equations with three or more variables. I would go with Cramer's Rule, but I guess that is against the point of the question. What you are going to do for elimination is pair up two out of the three equations and do an elimination making two variables stay. Then do the same for another pair of the three equations. Make sure that the variables in each of these two variable equations are the same. Now you can solve a normal linear system, and you will have two out of the three variables. Do you understand or would you like me to show you?
Alright. So i want to reduce this to a system of linear equations to solve this. Lets make an equation of x and y by using normal elimination with pairing equations 1 and 2.
Equations 1 and 2:
(-3x - 4y + 2z = 7)2
x - 3y + 4z = 5 -
-7x -5y = 9
Try to find another equation that will make a unique linear equation with the same variables as this one (Hint, use equations 2 and 3 and use elimination to create an x and y equation from them). Solve the system of linear equations by using the two ones you have made, plug x and y bag in to any of the three main equations to solve for z