1. **State the problem:** We are given a system of linear equations: $$3x - y = 1$$ $$7x + y = 9$$ We need to solve for $x$ and $y$. 2. **Choose a method:** We can use the elimination method because adding the two equations will eliminate $y$. 3. **Add the two equations:** $$ (3x - y) + (7x + y) = 1 + 9 $$ $$ 3x - y + 7x + y = 10 $$ $$ 10x = 10 $$ 4. **Solve for $x$:** $$ x = \frac{10}{10} $$ $$ x = 1 $$ 5. **Substitute $x=1$ into one of the original equations to find $y$:** Using $3x - y = 1$: $$ 3(1) - y = 1 $$ $$ 3 - y = 1 $$ $$ -y = 1 - 3 $$ $$ -y = -2 $$ $$ y = 2 $$ 6. **Final answer:** $$ x = 1, \quad y = 2 $$