The technique is usually to eliminate one variable between two equations, then eliminate the same variable using one of those equations and the remaining equation. So double the first equation and subtract the second equation from the doubled equation: 4y+5z=-16. Now triple the first equation and subtract the third equation: y+z=5.
Now we have two equations in y and z because x has been eliminated. We can use substitution: y=5-z and put this value of y in the other 2-variable equation: 4(5-z)+5z=-16; 20-4z+5z=-16, 20+z=-16, z=-36. Therefore y=5-z=41.
Finally use any of the equations to find x by substituting for y and z: x-41-36=4, so x=81.
Check the solution: 81-41-36=4; 162-246+108=24; 243-164-72=7. All OK. So x=81, y=41, z=-36.