My variable disappears when I'm solving this system of linear equations and I don't know what I'm doing wrong.

This is the only way I know how to solve this type of problem setup. Is there another method I can use to solve this?

Thanks

-10x + 30y + 10z = 0

5x - 40y + 5z = 0

5x + 10y - 15z = 0

x + y + z = 1

x = 1 - y - z

-10(1 - y - z) + 30y + 10z = 0

-10 + 10y + 10z + 30y + 10z = 0

-10 + 40y + 20z = 0

40y = -20z + 10

8y = -4x + 2

5(1 - y - z) - 40y + 5z = 0

5 -5y - 5z - 40y + 5z = 0

5 - 45y = 0 ??????