1. **State the problem:** We are given two equations: $$2x + \frac{1}{2}y = 21$$ $$y + \frac{4}{3}m = 18$$ We need to find the value of $x + y + m$. 2. **Express $y$ from the second equation:** $$y + \frac{4}{3}m = 18 \implies y = 18 - \frac{4}{3}m$$ 3. **Substitute $y$ into the first equation:** $$2x + \frac{1}{2} \left(18 - \frac{4}{3}m\right) = 21$$ 4. **Simplify the substitution:** $$2x + \frac{1}{2} \times 18 - \frac{1}{2} \times \frac{4}{3}m = 21$$ $$2x + 9 - \frac{2}{3}m = 21$$ 5. **Isolate $2x$:** $$2x = 21 - 9 + \frac{2}{3}m$$ $$2x = 12 + \frac{2}{3}m$$ 6. **Divide both sides by 2 to solve for $x$:** $$x = \frac{\cancel{2} \times 6 + \frac{2}{3}m}{\cancel{2}} = 6 + \frac{1}{3}m$$ 7. **Sum $x + y + m$ using expressions for $x$ and $y$:** $$x + y + m = \left(6 + \frac{1}{3}m\right) + \left(18 - \frac{4}{3}m\right) + m$$ 8. **Combine like terms:** $$= 6 + 18 + \frac{1}{3}m - \frac{4}{3}m + m$$ $$= 24 + \left(\frac{1}{3} - \frac{4}{3} + 1\right)m$$ 9. **Simplify the coefficients of $m$:** $$\frac{1}{3} - \frac{4}{3} + 1 = \frac{1 - 4 + 3}{3} = \frac{0}{3} = 0$$ 10. **Final result:** $$x + y + m = 24 + 0 = 24$$ **Answer:** $$\boxed{24}$$