1. **State the problem:** We are given three systems of linear equations (System A, B, and C) and need to classify each as consistent dependent, consistent independent, or inconsistent. Then, determine the solution type and find the solution if unique. 2. **Recall definitions:** - Consistent independent: exactly one unique solution (lines intersect at one point). - Consistent dependent: infinitely many solutions (lines coincide). - Inconsistent: no solution (parallel lines with different intercepts). 3. **Analyze System A:** Line 1: $y = -\frac{1}{2}x - \frac{3}{2}$ Line 2: $y = x - 3$ These lines have different slopes ($-\frac{1}{2}$ and $1$), so they intersect at exactly one point. 4. **Find intersection for System A:** Set $-\frac{1}{2}x - \frac{3}{2} = x - 3$ Multiply both sides by 2 to clear fraction: $$2\left(-\frac{1}{2}x - \frac{3}{2}\right) = 2(x - 3)$$ $$\cancel{2} \times -\frac{1}{2}x - \cancel{2} \times \frac{3}{2} = 2x - 6$$ $$-x - 3 = 2x - 6$$ Add $x$ to both sides: $$-x - 3 + x = 2x - 6 + x$$ $$-3 = 3x - 6$$ Add 6 to both sides: $$-3 + 6 = 3x - 6 + 6$$ $$3 = 3x$$ Divide both sides by 3: $$\frac{3}{\cancel{3}} = \frac{3x}{\cancel{3}}$$ $$1 = x$$ Substitute $x=1$ into Line 2: $$y = 1 - 3 = -2$$ Solution: $(1, -2)$ 5. **Analyze System B:** Line 1: $y = -\frac{2}{3}x - 2$ Line 2: $2x + 3y = -6$ Rewrite Line 2 in slope-intercept form: $$3y = -2x - 6$$ $$y = -\frac{2}{3}x - 2$$ Both lines have the same slope and intercept, so they are the same line. 6. **Classify System B:** Since both lines coincide, the system is consistent dependent with infinitely many solutions. 7. **Analyze System C:** Line 1: $y = \frac{3}{2}x - 1$ Line 2: $y = \frac{3}{2}x - 2$ Both lines have the same slope but different intercepts, so they are parallel and do not intersect. 8. **Classify System C:** The system is inconsistent with no solution. **Final classifications and solutions:** - System A: consistent independent, unique solution $(1, -2)$ - System B: consistent dependent, infinitely many solutions - System C: inconsistent, no solution