1. **State the problem:** Solve the system of equations by elimination: $$-v + w = 7$$ $$v + w = 1$$ 2. **Formula and rules:** The elimination method involves adding or subtracting equations to eliminate one variable, making it easier to solve for the other. 3. **Add the two equations to eliminate $v$:** $$(-v + w) + (v + w) = 7 + 1$$ $$\cancel{-v} + w + \cancel{v} + w = 8$$ $$2w = 8$$ 4. **Solve for $w$:** $$w = \frac{8}{2} = 4$$ 5. **Substitute $w=4$ into one of the original equations to find $v$:** Using $v + w = 1$: $$v + 4 = 1$$ $$v = 1 - 4 = -3$$ 6. **Check the solution in the other equation:** $$-v + w = -(-3) + 4 = 3 + 4 = 7$$ This matches the original equation, so the solution is correct. **Final answer:** $$v = -3, \quad w = 4$$