1. **State the problem:** Solve the system of linear equations: $$x + 4y = 5$$ $$3x + 5y = 1$$ 2. **Formula and method:** We can use the substitution or elimination method. Here, we'll 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 original equation to find $x$:** $$x + 4(2) = 5$$ $$x + 8 = 5$$ $$x = 5 - 8 = -3$$ 7. **Final answer:** $$x = -3, \quad y = 2$$