$$ \begin{aligned} 1000 &= \frac{1}{10}x+x&& \text{multiply ALL terms by } \color{blue}{ 10 }. \\[1 em]10\cdot1000 &= 10 \cdot \frac{1}{10}x+10x&& \text{cancel out the denominators} \\[1 em]10000 &= x+10x&& \text{simplify right side} \\[1 em]10000 &= 11x&& \text{ swap the left and right side of the equation } \\[1 em]11x &= 10000&& \text{ divide both sides by $ 11 $ } \\[1 em]x &= \frac{10000}{11}&& \\[1 em] \end{aligned} $$

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