1. **State the problem:** We have three systems of linear equations and need to understand their solutions and graph types. 2. **System 1:** $$-3x + y = 7$$ $$2x - 4y = -8$$ 3. **Solve System 1:** From the first equation, express $y$: $$y = 3x + 7$$ Substitute into the second: $$2x - 4(3x + 7) = -8$$ $$2x - 12x - 28 = -8$$ $$-10x - 28 = -8$$ $$-10x = 20$$ $$x = \frac{20}{-10} = -2$$ 4. Substitute $x = -2$ back to find $y$: $$y = 3(-2) + 7 = -6 + 7 = 1$$ 5. **Solution for System 1:** $$\boxed{(x,y) = (-2,1)}$$ This corresponds to the graph with exactly one intersection point. 6. **System 2:** $$3x - y = 4$$ $$6x - 2y = 8$$ 7. **Check if equations are multiples:** Multiply the first equation by 2: $$2(3x - y) = 2(4)$$ $$6x - 2y = 8$$ This matches the second equation exactly. 8. **Conclusion for System 2:** The two equations represent the same line, so there are infinitely many solutions (coincident lines). 9. **System 3:** $$y = -4x - 5$$ $$y = -4x + 1$$ 10. **Check slopes and intercepts:** Both lines have slope $-4$ but different intercepts ($-5$ and $1$). 11. **Conclusion for System 3:** Lines are parallel and never intersect, so there is no solution. **Summary:** - System 1: One solution (intersecting lines). - System 2: Infinitely many solutions (coincident lines). - System 3: No solution (parallel lines).