1. The problem asks about the slope formula applied to horizontal and vertical lines. The slope formula is $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line. - For a horizontal line, $y_2 = y_1$, so the numerator is zero. This means $$m = \frac{0}{x_2 - x_1} = 0$$. The slope of a horizontal line is always zero. - For a vertical line, $x_2 = x_1$, so the denominator is zero. This makes the slope formula undefined because division by zero is not allowed. Therefore, vertical lines have an undefined slope. 2. The problem gives the slope calculation for points $(2,7)$ and $(4,5)$: $$m = \frac{7 - 5}{4 - 2} = \frac{2}{2} = 1$$ Let's check carefully: $$7 - 5 = 2$$ is correct. $$4 - 2 = 2$$ is correct. So $$m = \frac{2}{2} = 1$$ is correct. There is no error in the calculation. The slope is 1. 3. Determine if lines are parallel, perpendicular, or neither. Recall: - Lines are parallel if their slopes are equal. - Lines are perpendicular if the product of their slopes is $-1$. (a) Line 1 points: $(1,0)$ and $(7,4)$ $$m_1 = \frac{4 - 0}{7 - 1} = \frac{4}{6} = \frac{2}{3}$$ Line 2 points: $(7,0)$ and $(3,6)$ $$m_2 = \frac{6 - 0}{3 - 7} = \frac{6}{-4} = -\frac{3}{2}$$ Check if parallel: $\frac{2}{3} \neq -\frac{3}{2}$ Check if perpendicular: $$\frac{2}{3} \times -\frac{3}{2} = -1$$ Since product is $-1$, lines are perpendicular. (b) Line 1 points: $(3,5)$ and $(5,10)$ $$m_1 = \frac{10 - 5}{5 - 3} = \frac{5}{2}$$ Line 2 points: $(1,-3)$ and $(7,12)$ $$m_2 = \frac{12 - (-3)}{7 - 1} = \frac{15}{6} = \frac{5}{2}$$ Slopes are equal, so lines are parallel. 4. Graph a line through $(0,2)$ parallel to the line through $(-2,4)$ and $(-5,1)$. First, find the slope of the given line: $$m = \frac{1 - 4}{-5 - (-2)} = \frac{-3}{-3} = 1$$ Since the new line is parallel, it has the same slope $m=1$. Using point-slope form with point $(0,2)$: $$y - 2 = 1(x - 0)$$ Simplify: $$y = x + 2$$ This is the equation of the line to graph. Final answers: 1. Horizontal lines have slope 0; vertical lines have undefined slope. 2. The slope calculation is correct; slope is 1. 3a. Lines are perpendicular. 3b. Lines are parallel. 4. Equation of the line is $$y = x + 2$$.