1. **Problem Statement:** Classify the system of linear equations and find the solution if it exists. --- ### System A Equations: $$y = x + 1$$ $$y = -x + 1$$ 2. **Classification:** These are two lines with different slopes ($1$ and $-1$), so they intersect at exactly one point. 3. **Find the solution:** Set the right sides equal since both equal $y$: $$x + 1 = -x + 1$$ 4. **Solve for $x$:** $$x + 1 = -x + 1$$ $$x + x = 1 - 1$$ $$2x = 0$$ $$x = 0$$ 5. **Find $y$:** Substitute $x=0$ into $y = x + 1$: $$y = 0 + 1 = 1$$ 6. **Solution:** $(0, 1)$ 7. **Classification:** The system is **consistent independent** (one unique solution). --- ### System B Equations: $$y = -x + 3$$ $$x + y = 3$$ 8. **Rewrite second equation:** $$y = 3 - x$$ 9. **Compare both equations:** $$y = -x + 3$$ $$y = 3 - x$$ They are the same equation, so the lines coincide. 10. **Classification:** The system is **consistent dependent** (infinitely many solutions). 11. **Solution:** Infinitely many solutions along the line $y = -x + 3$. --- ### System C Equations: $$y = \frac{3}{2}x + 2$$ $$y = \frac{3}{2}x$$ 12. **Classification:** Both lines have the same slope $\frac{3}{2}$ but different intercepts ($2$ and $0$), so they are parallel and do not intersect. 13. **Classification:** The system is **inconsistent** (no solution). --- **Final answers:** - System A: consistent independent, unique solution $(0,1)$ - System B: consistent dependent, infinitely many solutions - System C: inconsistent, no solution