1. **State the problem:** We are given a vector expression in terms of variables $x_0$, $x_5$, and $x_7$: $$\begin{bmatrix} y \\ z \\ w \end{bmatrix} = \begin{bmatrix} -\frac{1}{2} x_0 + \frac{4}{3} x_5 - \frac{1}{2} x_7 \\ \frac{1}{20} x_0 - \frac{3}{20} x_5 + \frac{1}{4} x_7 \\ \frac{3}{20} x_0 + \frac{20}{5} x_5 + \frac{1}{4} x_7 \end{bmatrix}$$ We want to understand or simplify this vector expression. 2. **Analyze each component:** - First component: $$-\frac{1}{2} x_0 + \frac{4}{3} x_5 - \frac{1}{2} x_7$$ - Second component: $$\frac{1}{20} x_0 - \frac{3}{20} x_5 + \frac{1}{4} x_7$$ - Third component: $$\frac{3}{20} x_0 + \frac{20}{5} x_5 + \frac{1}{4} x_7$$ 3. **Simplify fractions where possible:** - Note that $$\frac{20}{5} = 4$$ So the third component becomes: $$\frac{3}{20} x_0 + 4 x_5 + \frac{1}{4} x_7$$ 4. **Final simplified vector expression:** $$\begin{bmatrix} -\frac{1}{2} x_0 + \frac{4}{3} x_5 - \frac{1}{2} x_7 \\ \frac{1}{20} x_0 - \frac{3}{20} x_5 + \frac{1}{4} x_7 \\ \frac{3}{20} x_0 + 4 x_5 + \frac{1}{4} x_7 \end{bmatrix}$$ This is the simplified form of the vector expression in terms of $x_0$, $x_5$, and $x_7$. No further simplification is possible without specific values for $x_0$, $x_5$, and $x_7$. **Answer:** $$\boxed{\begin{bmatrix} -\frac{1}{2} x_0 + \frac{4}{3} x_5 - \frac{1}{2} x_7 \\ \frac{1}{20} x_0 - \frac{3}{20} x_5 + \frac{1}{4} x_7 \\ \frac{3}{20} x_0 + 4 x_5 + \frac{1}{4} x_7 \end{bmatrix}}$$