1. **Problem:** Solve the system of equations: $$x + 4y = 5$$ $$3x + 5y = 1$$ 2. **Formula and rules:** To solve a system of linear equations, we can use substitution or elimination. Here, we use elimination. 3. **Step 1:** Multiply the first equation by 3 to align coefficients of $x$: $$3(x + 4y) = 3(5) \Rightarrow 3x + 12y = 15$$ 4. **Step 2:** Subtract the second equation from this new equation: $$ (3x + 12y) - (3x + 5y) = 15 - 1 $$ $$ 3x - 3x + 12y - 5y = 14 $$ $$ 7y = 14 $$ 5. **Step 3:** Solve for $y$: $$ y = \frac{14}{7} = 2 $$ 6. **Step 4:** Substitute $y=2$ into the first equation: $$ x + 4(2) = 5 $$ $$ x + 8 = 5 $$ $$ x = 5 - 8 = -3 $$ 7. **Answer:** The solution to the system is: $$ x = -3, \quad y = 2 $$