1. **State the problem:** Solve the system of linear equations for $i$ and $g$ given: $$\begin{cases} 5i + 2g = 5020 \\ 3i + 8g = 7840 \end{cases}$$ 2. **Explain the method:** We will use substitution or elimination to find $i$ and $g$. Here, substitution is convenient since $g$ was found in the user's message as $g=710$. 3. **Substitute $g=710$ into the first equation:** $$5i + 2(710) = 5020$$ Simplify: $$5i + 1420 = 5020$$ 4. **Isolate $i$:** $$5i = 5020 - 1420$$ $$5i = 3600$$ 5. **Solve for $i$:** $$i = \frac{3600}{5}$$ Show cancellation: $$i = \frac{\cancel{3600}}{\cancel{5}} = 720$$ 6. **Verify with the second equation:** $$3i + 8g = 7840$$ Substitute $i=720$ and $g=710$: $$3(720) + 8(710) = 2160 + 5680 = 7840$$ This confirms the solution. **Final answer:** $$i = 720, \quad g = 710$$