1. **State the problem:** Determine the relationship between the lines and find the number of solutions for the system of equations. 2. **Recall the slope-intercept form:** The equation of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. 3. **Analyze the first pair:** - $y = -3x + 1$ has slope $m_1 = -3$. - $y = 3x + 1$ has slope $m_2 = 3$. Since $m_1 \neq m_2$, the lines are not parallel. However, the slopes are not equal, so the lines are not parallel but intersecting. But the user states "Parallel" and "Number of Solutions: 0" which is incorrect because slopes differ. 4. **Analyze the second pair:** - $y = \frac{4}{3}x - 2$ and $y = \frac{4}{3}x + 4$ have the same slope $\frac{4}{3}$ but different y-intercepts. - Lines with the same slope but different intercepts are parallel. - So, the relationship is Parallel, and number of solutions is 0. 5. **Analyze the third pair:** - $x - 3y = 7$ and $4x + 2y = 14$. - Convert to slope-intercept form: $$x - 3y = 7 \Rightarrow -3y = 7 - x \Rightarrow y = \frac{x}{3} - \frac{7}{3}$$ $$4x + 2y = 14 \Rightarrow 2y = 14 - 4x \Rightarrow y = 7 - 2x$$ - Slopes are $\frac{1}{3}$ and $-2$, which are different, so lines intersect. 6. **Analyze the fourth pair:** - $y = \frac{3}{2}x + 1$ and $y = \frac{1}{2}(3x - 8) + 2$. - Simplify second equation: $$y = \frac{1}{2}(3x - 8) + 2 = \frac{3}{2}x - 4 + 2 = \frac{3}{2}x - 2$$ - Both lines have slope $\frac{3}{2}$ but different intercepts. - Lines are parallel, no solution. - However, user states solution is $(-6, -9)$ which is incorrect for parallel lines. 7. **Analyze the fifth pair:** - $y = -2x + 2$ and $y = \frac{5}{2}x - 7$. - Slopes are $-2$ and $\frac{5}{2}$, different slopes, so lines intersect. - User states solution is $(2, -2)$. 8. **Verify solution for fifth pair:** - Substitute $x=2$ into first equation: $$y = -2(2) + 2 = -4 + 2 = -2$$ - Substitute $x=2$ into second equation: $$y = \frac{5}{2}(2) - 7 = 5 - 7 = -2$$ - Both give $y = -2$, so solution $(2, -2)$ is correct. **Final answer:** - First pair: Slopes differ, lines intersect, 1 solution. - Second pair: Same slope, different intercepts, parallel, 0 solutions. - Third pair: Different slopes, intersect, 1 solution. - Fourth pair: Same slope, different intercepts, parallel, 0 solutions (user's solution incorrect). - Fifth pair: Different slopes, intersect, 1 solution at $(2, -2)$.