1. **State the problem:** Solve the system of linear equations: $$-6x + y = 3$$ $$2x + 4y = 1$$ 2. **Formula and rules:** We can solve this system using substitution or elimination. Here, we'll use elimination. 3. **Step 1: Multiply the first equation by 4 to align coefficients of $y$:** $$4(-6x + y) = 4(3) \Rightarrow -24x + 4y = 12$$ 4. **Step 2: Write the system now:** $$-24x + 4y = 12$$ $$2x + 4y = 1$$ 5. **Step 3: Subtract the second equation from the first to eliminate $y$:** $$(-24x + 4y) - (2x + 4y) = 12 - 1$$ $$-24x + 4y - 2x - 4y = 11$$ $$-26x = 11$$ 6. **Step 4: Solve for $x$:** $$x = \frac{11}{-26} = -\frac{11}{26}$$ 7. **Step 5: Substitute $x$ back into the first original equation to find $y$:** $$-6\left(-\frac{11}{26}\right) + y = 3$$ $$\frac{66}{26} + y = 3$$ 8. **Step 6: Simplify and solve for $y$:** $$y = 3 - \frac{66}{26} = \frac{78}{26} - \frac{66}{26} = \frac{12}{26} = \frac{6}{13}$$ **Final answer:** $$x = -\frac{11}{26}, \quad y = \frac{6}{13}$$