1. **Problem statement:** We are given a vector $r = \begin{pmatrix}80 \\ 50\end{pmatrix}$ and a vector $x = \begin{pmatrix}30 \\ 20\end{pmatrix}$. We need to analyze these vectors and interpret the solution in the context of the problem. 2. **Step 1: Understanding vectors** Vectors are quantities with both magnitude and direction, represented here as column matrices. 3. **Step 2: Check components of $x$** The vector $x$ has components 30 and 20, both positive. 4. **Step 3: Interpretation of negative components** If $x$ had a negative component, it would mean a negative quantity in that dimension, which might not make sense in the real-world context (e.g., negative amount of raw material). 5. **Step 4: Contextual constraint** The problem states that solutions with negative components are not valid in the real-world context, for example, if not all raw material is used up. 6. **Final answer:** The vector $x$ is valid as both components are positive. If $x$ had negative components, it would not be a valid solution in the given context.