jmc

To start, find a common denominator on the left side. The lowest common multiple of 7 and 3 is 21, so this is the common denominator. Rewrite the equation as: $$\begin{array}{rl}\frac{3x}{21}+\frac{7x}{21}& =1-x\\ \frac{10x}{21}& =1-x\end{array}$$We now multiply both sides by $21$ to get $10x=21-21x,$ then $31x=21.$ So $x=\boxed{\frac{21}{31}}.$