1. **State the problem:** Solve the system of linear equations: $$5x - y = 18 \quad (1)$$ $$2x + 3y = 31 \quad (2)$$ 2. **Formula and method:** We can use substitution or elimination to solve for $x$ and $y$. Here, we'll use substitution. 3. **Isolate $y$ in equation (1):** $$5x - y = 18 \implies y = 5x - 18$$ 4. **Substitute $y$ into equation (2):** $$2x + 3(5x - 18) = 31$$ 5. **Simplify and solve for $x$:** $$2x + 15x - 54 = 31$$ $$17x - 54 = 31$$ $$17x = 31 + 54$$ $$17x = 85$$ $$x = \frac{85}{17} = 5$$ 6. **Find $y$ using $x=5$ in $y = 5x - 18$:** $$y = 5(5) - 18 = 25 - 18 = 7$$ 7. **Final answer:** $$x = 5, \quad y = 7$$ This means the solution to the system is the point $(5,7)$ where both lines intersect.