1. **State the problem:** Classify each system of linear equations as consistent dependent, consistent independent, or inconsistent, and find the solution if it exists. --- ### System A 2. **Equations:** $$y = -\frac{1}{2}x - 3$$ $$y = -\frac{1}{2}x - 2$$ 3. **Analyze slopes and intercepts:** Both lines have the same slope $-\frac{1}{2}$ but different y-intercepts ($-3$ and $-2$). 4. **Interpretation:** Lines with the same slope but different intercepts are parallel and never intersect. 5. **Conclusion:** The system is **inconsistent**. 6. **Solution:** No solution. --- ### System B 7. **Equations:** $$y = \frac{3}{2}x + 3$$ $$-3x + 2y = 6$$ 8. **Rewrite second equation in slope-intercept form:** $$-3x + 2y = 6 \implies 2y = 3x + 6 \implies y = \frac{3}{2}x + 3$$ 9. **Compare equations:** Both lines are identical. 10. **Interpretation:** The system is **consistent dependent**. 11. **Solution:** Infinitely many solutions (all points on the line). --- ### System C 12. **Equations:** $$y = -x + 1$$ $$y = \frac{1}{2}x - \frac{7}{2}$$ 13. **Set equations equal to find intersection:** $$-x + 1 = \frac{1}{2}x - \frac{7}{2}$$ 14. **Solve for $x$:** $$-x - \frac{1}{2}x = -\frac{7}{2} - 1$$ $$-\frac{3}{2}x = -\frac{9}{2}$$ 15. **Divide both sides by $-\frac{3}{2}$:** $$x = \frac{-\frac{9}{2}}{-\frac{3}{2}} = \cancel{\frac{9}{2}} \times \frac{2}{3} = 3$$ 16. **Find $y$ by substituting $x=3$ into first equation:** $$y = -3 + 1 = -2$$ 17. **Conclusion:** The system is **consistent independent** with a unique solution. 18. **Solution:** $(3, -2)$ --- **Final answers:** - System A: inconsistent, no solution. - System B: consistent dependent, infinitely many solutions. - System C: consistent independent, unique solution $(3, -2)$.