1. **State the problem:** Solve the system of linear equations: $$5a + 2 + b = 15$$ $$2a + b = 5$$ 2. **Rewrite the first equation:** Subtract 2 from both sides to isolate terms with variables: $$5a + b + 2 = 15 \implies 5a + b = 15 - 2$$ $$5a + b = 13$$ 3. **Write the system clearly:** $$\begin{cases} 5a + b = 13 \\ 2a + b = 5 \end{cases}$$ 4. **Subtract the second equation from the first to eliminate $b$:** $$ (5a + b) - (2a + b) = 13 - 5 $$ $$ 5a + b - 2a - b = 8 $$ $$ 3a = 8 $$ 5. **Solve for $a$:** $$ a = \frac{8}{3} $$ 6. **Substitute $a = \frac{8}{3}$ into the second equation to find $b$:** $$ 2\left(\frac{8}{3}\right) + b = 5 $$ $$ \frac{16}{3} + b = 5 $$ 7. **Isolate $b$:** $$ b = 5 - \frac{16}{3} $$ $$ b = \frac{15}{3} - \frac{16}{3} $$ $$ b = \frac{15 - 16}{3} $$ $$ b = -\frac{1}{3} $$ **Final answer:** $$ a = \frac{8}{3}, \quad b = -\frac{1}{3} $$