1. **State the problem:** Solve the system of equations using elimination: $$\begin{cases} 2x + 3y = 8 \\ 5x + 6y = 17 \end{cases}$$ 2. **Explain the elimination method:** The goal is to eliminate one variable by making the coefficients of that variable equal (or opposites) in both equations, then subtract or add the equations. 3. **Make coefficients of $y$ equal:** Multiply the first equation by 2 to match the coefficient of $y$ in the second equation: $$2 \times (2x + 3y) = 2 \times 8 \Rightarrow 4x + 6y = 16$$ 4. **Subtract the second equation from this new equation:** $$ (4x + 6y) - (5x + 6y) = 16 - 17 $$ Simplify: $$4x - 5x + 6y - 6y = -1$$ $$-x = -1$$ 5. **Solve for $x$:** $$x = 1$$ 6. **Substitute $x=1$ into the first original equation:** $$2(1) + 3y = 8$$ $$2 + 3y = 8$$ 7. **Solve for $y$:** $$3y = 8 - 2 = 6$$ $$y = 2$$ 8. **Final answer:** $$\boxed{(x, y) = (1, 2)}$$ This is the point where the two lines intersect, confirming the solution to the system.