1. **State the problem:** We are given the system of equations: $$x + y = 10$$ $$x = 2y + 1$$ 2. **Goal:** Find the values of $x$ and $y$ that satisfy both equations simultaneously. 3. **Substitution method:** Since $x$ is expressed in terms of $y$ in the second equation, substitute $x = 2y + 1$ into the first equation: $$ (2y + 1) + y = 10 $$ 4. **Simplify the equation:** $$ 2y + 1 + y = 10 $$ $$ 3y + 1 = 10 $$ 5. **Isolate $y$:** $$ 3y = 10 - 1 $$ $$ 3y = 9 $$ 6. **Divide both sides by 3:** $$ y = \frac{\cancel{3}y}{\cancel{3}} = \frac{9}{3} $$ $$ y = 3 $$ 7. **Find $x$ using $x = 2y + 1$:** $$ x = 2(3) + 1 $$ $$ x = 6 + 1 $$ $$ x = 7 $$ **Final answer:** $$ x = 7, \quad y = 3 $$