1. **State the problem:** We have two equations based on Alfie's and Lyra's operations with numbers $c$ and $d$: - Alfie triples $c$ and adds two lots of $d$ to get 35. - Lyra multiplies $c$ by five and adds four lots of $d$ to get 61. 2. **Write the equations:** From the problem, we get: $$3c + 2d = 35$$ $$5c + 4d = 61$$ 3. **Solve the system of equations:** We can use the method of elimination or substitution. Here, we'll use elimination. Multiply the first equation by 2 to align the coefficients of $d$: $$2(3c + 2d) = 2(35)$$ $$6c + 4d = 70$$ Now subtract the second equation from this new equation: $$\cancel{6c} + 4d - (5c + 4d) = 70 - 61$$ $$6c + 4d - 5c - 4d = 9$$ $$c = 9$$ 4. **Substitute $c=9$ back into one of the original equations:** Using the first equation: $$3(9) + 2d = 35$$ $$27 + 2d = 35$$ Subtract 27 from both sides: $$2d = 35 - 27$$ $$2d = 8$$ Divide both sides by 2: $$\cancel{2}d / \cancel{2} = 8 / 2$$ $$d = 4$$ 5. **Final answer:** $$c = 9, \quad d = 4$$