1. **State the problem:** Solve the linear system by graphing and checking the solution for the first system: $$y = -x + 3$$ $$y = x + 1$$ 2. **Understand the system:** We have two linear equations. The solution is the point where the two lines intersect. 3. **Set the equations equal to find the intersection:** Since both equal $y$, set: $$-x + 3 = x + 1$$ 4. **Solve for $x$:** $$-x + 3 = x + 1$$ $$3 - 1 = x + x$$ $$2 = 2x$$ $$x = 1$$ 5. **Find $y$ by substituting $x=1$ into one of the equations:** Using $y = x + 1$: $$y = 1 + 1 = 2$$ 6. **Check the solution in the other equation:** Using $y = -x + 3$: $$y = -1 + 3 = 2$$ Both equations give $y=2$, so the solution is correct. **Final answer:** The solution to the system is $$\boxed{(1, 2)}$$.