1. **State the problem:** Solve the system of complex linear equations: $$2z + w = 4 + i$$ $$6z - w = 8 + 2i$$ 2. **Formula and rules:** We can solve this system by adding or subtracting equations to eliminate one variable, then solve for the other. 3. **Add the two equations to eliminate $w$:** $$ (2z + w) + (6z - w) = (4 + i) + (8 + 2i) $$ $$ 2z + w + 6z - w = 12 + 3i $$ $$ 8z = 12 + 3i $$ 4. **Solve for $z$:** $$ z = \frac{12 + 3i}{8} = \frac{12}{8} + \frac{3i}{8} = 1.5 + 0.375i $$ 5. **Substitute $z$ back into the first equation to find $w$:** $$ 2z + w = 4 + i $$ $$ 2(1.5 + 0.375i) + w = 4 + i $$ $$ 3 + 0.75i + w = 4 + i $$ $$ w = (4 + i) - (3 + 0.75i) = 1 + 0.25i $$ 6. **Final answer:** $$ z = 1.5 + 0.375i, \quad w = 1 + 0.25i $$