1. **Identify the slope and y-intercept for the equation** $y = - \frac{5}{6} x + 4$. The slope-intercept form is $y = mx + b$ where $m$ is the slope and $b$ is the y-intercept. Here, $m = - \frac{5}{6}$ and $b = 4$. 2. **Identify the slope and y-intercept for** $f(x) = 3x - 7$. Rewrite as $y = 3x - 7$. Slope $m = 3$, y-intercept $b = -7$. 3. **Rewrite** $x - y = 8$ **in slope-intercept form**. Subtract $x$ from both sides: $$-y = -x + 8$$ Multiply both sides by $-1$: $$\cancel{-}y = \cancel{-}(-x + 8)$$ $$y = x - 8$$ Slope $m = 1$, y-intercept $b = -8$. 4. **Rewrite** $12x + 8y = 28$ **in slope-intercept form**. Subtract $12x$ from both sides: $$8y = -12x + 28$$ Divide both sides by $8$: $$y = \frac{\cancel{8}y}{\cancel{8}} = \frac{-12x + 28}{8} = -\frac{12}{8}x + \frac{28}{8}$$ Simplify fractions: $$y = -\frac{3}{2}x + \frac{7}{2}$$ Slope $m = -\frac{3}{2}$, y-intercept $b = \frac{7}{2}$. 5. **Rewrite** $x - 2y - 2 = 0$ **in slope-intercept form**. Add $2$ to both sides: $$x - 2y = 2$$ Subtract $x$ from both sides: $$-2y = -x + 2$$ Divide both sides by $-2$: $$y = \frac{\cancel{-2}y}{\cancel{-2}} = \frac{-x + 2}{-2} = \frac{-x}{-2} + \frac{2}{-2} = \frac{1}{2}x - 1$$ Slope $m = \frac{1}{2}$, y-intercept $b = -1$. 6. **Rewrite** $-5x = 2y$ **in slope-intercept form**. Divide both sides by $2$: $$y = \frac{-5x}{2} = -\frac{5}{2}x$$ Slope $m = -\frac{5}{2}$, y-intercept $b = 0$. 7. **Write** $y = 6x - 4$ **in standard form**. Subtract $6x$ from both sides: $$-6x + y = -4$$ Multiply both sides by $-1$ to make $x$ coefficient positive: $$6x - y = 4$$ Standard form: $6x - y = 4$. 8. **Write** $5x + 15y = 75$ **in standard form**. This is already in standard form. 9. **Rewrite** $4x - 8 = 9y - 5$ **in standard form**. Add $8$ to both sides: $$4x = 9y - 5 + 8$$ $$4x = 9y + 3$$ Subtract $9y$ from both sides: $$4x - 9y = 3$$ Standard form: $4x - 9y = 3$. 10. **Rewrite** $\frac{3}{5} x + \frac{1}{4} y = - \frac{1}{2}$ **in standard form**. Multiply entire equation by the least common denominator $20$: $$20 \times \left( \frac{3}{5} x + \frac{1}{4} y \right) = 20 \times \left(- \frac{1}{2} \right)$$ $$20 \times \frac{3}{5} x + 20 \times \frac{1}{4} y = -10$$ $$12x + 5y = -10$$ Standard form: $12x + 5y = -10$. **Final answers:** 1. $m = -\frac{5}{6}$, $b = 4$ 2. $m = 3$, $b = -7$ 3. $m = 1$, $b = -8$ 4. $m = -\frac{3}{2}$, $b = \frac{7}{2}$ 5. $m = \frac{1}{2}$, $b = -1$ 6. $m = -\frac{5}{2}$, $b = 0$ 7. $6x - y = 4$ 8. $5x + 15y = 75$ 9. $4x - 9y = 3$ 10. $12x + 5y = -10$