1. **State the problem:** We are given three systems of linear equations (System A, B, and C). We need to classify each system as "consistent dependent," "consistent independent," or "inconsistent." Then, determine the nature of their solutions and find the solution if it is unique. 2. **Recall definitions:** - A system is **consistent independent** if it has exactly one unique solution (lines intersect at one point). - A system is **consistent dependent** if it has infinitely many solutions (lines coincide). - A system is **inconsistent** if it has no solution (lines are parallel and distinct). --- ### System A: Line 1: $y = -\frac{1}{2}x + \frac{7}{2}$ Line 2: $y = -x + 5$ 3. **Check slopes and intercepts:** - Slope of Line 1: $-\frac{1}{2}$ - Slope of Line 2: $-1$ Since slopes are different, lines intersect at exactly one point. 4. **Find intersection point:** Set the right sides equal: $$-\frac{1}{2}x + \frac{7}{2} = -x + 5$$ 5. **Solve for $x$:** $$-\frac{1}{2}x + \frac{7}{2} = -x + 5$$ Add $x$ to both sides: $$-\frac{1}{2}x + x + \frac{7}{2} = 5$$ Simplify: $$\frac{1}{2}x + \frac{7}{2} = 5$$ Subtract $\frac{7}{2}$ from both sides: $$\frac{1}{2}x = 5 - \frac{7}{2} = \frac{10}{2} - \frac{7}{2} = \frac{3}{2}$$ Multiply both sides by 2: $$\cancel{2} \times \frac{1}{2}x = \cancel{2} \times \frac{3}{2}$$ $$x = 3$$ 6. **Find $y$:** Substitute $x=3$ into Line 1: $$y = -\frac{1}{2} \times 3 + \frac{7}{2} = -\frac{3}{2} + \frac{7}{2} = \frac{4}{2} = 2$$ 7. **Conclusion for System A:** - Classification: **consistent independent** - Solution: $(3, 2)$ --- ### System B: Line 1: $y = -1$ Line 2: $y = 4$ 8. **Check slopes:** Both lines are horizontal with slopes 0 but different $y$-intercepts. 9. **Since lines are parallel and distinct, no intersection exists.** 10. **Conclusion for System B:** - Classification: **inconsistent** - Solution: **no solution** --- ### System C: Line 1: $y = \frac{2}{3}x - 3$ Line 2: $-2x + 3y = -9$ 11. **Rewrite Line 2 in slope-intercept form:** $$3y = 2x - 9$$ $$y = \frac{2}{3}x - 3$$ 12. **Both lines have the same equation, so they coincide.** 13. **Conclusion for System C:** - Classification: **consistent dependent** - Solution: **infinitely many solutions** --- **Final answers:** - System A: consistent independent, unique solution $(3, 2)$ - System B: inconsistent, no solution - System C: consistent dependent, infinitely many solutions