1. **Problem:** Find the slope of each line segment given rise and run values. **Formula:** Slope $m = \frac{\text{rise}}{\text{run}}$ **Rules:** Simplify fractions to lowest terms. If run is negative, simplify sign accordingly. **Solutions:** - a) $m = \frac{2}{3}$ (already simplified) - b) $m = \frac{2}{8} = \frac{1}{4}$ - c) $m = \frac{-5}{7}$ (already simplified) - d) $m = \frac{6}{-18} = -\frac{1}{3} = -3$ (since $-\frac{1}{3}$ is $-0.333$, but answer given is $-3$, so likely a typo; correct slope is $-\frac{1}{3}$) 2. **Problem:** Determine slope as decimal given rise and run. **Formula:** $m = \frac{\text{rise}}{\text{run}}$ **Solutions:** - a) $m = \frac{3}{10} = 0.3$ - b) $m = \frac{-9}{8} = -1.125$ - c) $m = \frac{-4.25}{0.5} = -8.5$ - d) $m = \frac{0}{17} = 0$ 4. **Problem:** Determine slope from points on line graphs. **Formula:** $m = \frac{y_2 - y_1}{x_2 - x_1}$ **Solutions:** - a) From $(0,0)$ to $(7,3)$: $m = \frac{3-0}{7-0} = \frac{3}{7}$ - b) From $(0,9)$ to $(7,0)$: $m = \frac{0-9}{7-0} = -\frac{9}{7}$ - c) From $(0,2)$ to $(7,7)$: $m = \frac{7-2}{7-0} = \frac{5}{7}$ - d) Same as b): $m = -\frac{9}{7}$ 7. **Problem:** Determine slope from given points. **Solutions:** - a) $(0,2)$ to $(10,14)$: $m = \frac{14-2}{10-0} = \frac{12}{10} = 1.2$ - b) $(0,20)$ to $(20,92)$: $m = \frac{92-20}{20-0} = \frac{72}{20} = 3.6$ - c) $(0,900)$ to $(40,0)$: $m = \frac{0-900}{40-0} = \frac{-900}{40} = -22.5$ (answer given is -25, likely rounding or typo; correct is -22.5) 8. **Problem:** Determine slope from line descriptions. **Solutions:** - a) Rising line in positive quadrant: $m=3$ - b) Falling line in positive/negative quadrants: $m=-2$ - c) Rising from $x=-16$ to $x=16$: $m=1$ - d) Horizontal line $y=4000$: $m=0$ - e) Falling from positive to negative $x,y$: $m=-4$ - f) Vertical line $x=0$: slope is undefined 9. **Problem:** Determine slope from pairs of points. **Formula:** $m = \frac{y_2 - y_1}{x_2 - x_1}$ **Solutions:** - a) $(2,4)$ and $(5,10)$: $m = \frac{10-4}{5-2} = \frac{6}{3} = 2$ - b) $(5,7)$ and $(8,-2)$: $m = \frac{-2-7}{8-5} = \frac{-9}{3} = -3$ - c) $(-1,0)$ and $(12,5)$: $m = \frac{5-0}{12+1} = \frac{5}{13} \approx 0.3846$ (answer given $5/11$ likely typo; correct is $5/13$) - d) $(5,17)$ and $(8,5)$: $m = \frac{5-17}{8-5} = \frac{-12}{3} = -4$ - e) $(10,-3)$ and $(-14,-21)$: $m = \frac{-21+3}{-14-10} = \frac{-18}{-24} = \frac{3}{4}$ - f) $(15,17)$ and $(9,17)$: $m = \frac{17-17}{9-15} = 0$ - g) $(0.5,-2)$ and $(0,4.5)$: $m = \frac{4.5+2}{0-0.5} = \frac{6.5}{-0.5} = -13$ (answer given 13, sign reversed) - h) $(-40,54)$ and $(-40,38)$: vertical line, slope undefined - i) $(-1.8,1.4)$ and $(3.4,2.5)$: $m = \frac{2.5-1.4}{3.4+1.8} = \frac{1.1}{5.2} \approx 0.2115$ (answer given 0.25, approximate) **Final note:** Some answers in the provided box have minor inconsistencies; the above steps show correct calculations.