1. **State the problem:** Solve the system of linear equations: $$2x + 2y + 2z = 15$$ $$8x - 4y + 2z = 9$$ 2. **Identify the system:** We have two equations with three variables $x$, $y$, and $z$. This typically means infinite solutions along a line or plane unless additional constraints are given. 3. **Simplify the first equation:** Divide the entire first equation by 2: $$x + y + z = \frac{15}{2} = 7.5$$ 4. **Rewrite the system:** $$x + y + z = 7.5$$ $$8x - 4y + 2z = 9$$ 5. **Express $z$ from the first equation:** $$z = 7.5 - x - y$$ 6. **Substitute $z$ into the second equation:** $$8x - 4y + 2(7.5 - x - y) = 9$$ 7. **Simplify:** $$8x - 4y + 15 - 2x - 2y = 9$$ $$6x - 6y + 15 = 9$$ 8. **Isolate terms:** $$6x - 6y = 9 - 15$$ $$6x - 6y = -6$$ 9. **Divide entire equation by 6:** $$x - y = -1$$ 10. **Express $x$ in terms of $y$:** $$x = y - 1$$ 11. **Substitute $x$ back into $z$ equation:** $$z = 7.5 - (y - 1) - y = 7.5 - y + 1 - y = 8.5 - 2y$$ 12. **Final parametric solution:** $$x = y - 1$$ $$z = 8.5 - 2y$$ where $y$ is a free parameter. **Answer:** The system has infinitely many solutions parameterized by $y$: $$\boxed{(x, y, z) = (y - 1, y, 8.5 - 2y)}$$