1. Solve the equation $$\frac{1}{v} + \frac{3v + 12}{v^2 - 5v} = \frac{7v - 56}{v^2 - 5v}$$ - First, factor the denominators: $$v^2 - 5v = v(v - 5)$$. - Rewrite the equation with common denominators: $$\frac{1}{v} + \frac{3v + 12}{v(v - 5)} = \frac{7v - 56}{v(v - 5)}$$. - Multiply the first term by $$\frac{v - 5}{v - 5}$$ to get a common denominator: $$\frac{v - 5}{v(v - 5)} + \frac{3v + 12}{v(v - 5)} = \frac{7v - 56}{v(v - 5)}$$. - Combine the left side: $$\frac{v - 5 + 3v + 12}{v(v - 5)} = \frac{7v - 56}{v(v - 5)}$$. - Simplify numerator: $$\frac{4v + 7}{v(v - 5)} = \frac{7v - 56}{v(v - 5)}$$. - Since denominators are equal, set numerators equal: $$4v + 7 = 7v - 56$$. - Solve for $$v$$: $$4v + 7 = 7v - 56 \Rightarrow 7 + 56 = 7v - 4v \Rightarrow 63 = 3v \Rightarrow v = 21$$. 2. Solve the equation $$\frac{1}{m^2 - m} + \frac{1}{m} = \frac{5}{m^2 - m}$$ - Factor denominator: $$m^2 - m = m(m - 1)$$. - Rewrite equation: $$\frac{1}{m(m - 1)} + \frac{1}{m} = \frac{5}{m(m - 1)}$$. - Multiply second term by $$\frac{m - 1}{m - 1}$$: $$\frac{1}{m(m - 1)} + \frac{m - 1}{m(m - 1)} = \frac{5}{m(m - 1)}$$. - Combine left side: $$\frac{1 + m - 1}{m(m - 1)} = \frac{5}{m(m - 1)}$$. - Simplify numerator: $$\frac{m}{m(m - 1)} = \frac{5}{m(m - 1)}$$. - Set numerators equal: $$m = 5$$. 3. Solve the equation $$\frac{1}{n - 8} - 1 = \frac{7}{n - 8}$$ - Rewrite $$-1$$ as $$\frac{n - 8}{n - 8}$$: $$\frac{1}{n - 8} - \frac{n - 8}{n - 8} = \frac{7}{n - 8}$$. - Combine left side: $$\frac{1 - (n - 8)}{n - 8} = \frac{7}{n - 8}$$. - Simplify numerator: $$\frac{1 - n + 8}{n - 8} = \frac{7}{n - 8} \Rightarrow \frac{9 - n}{n - 8} = \frac{7}{n - 8}$$. - Set numerators equal: $$9 - n = 7 \Rightarrow 9 - 7 = n \Rightarrow n = 2$$. 4. Solve the equation $$\frac{1}{r - 2} + \frac{1}{r^2 - 7r + 10} = \frac{6}{r - 2}$$ - Factor denominator: $$r^2 - 7r + 10 = (r - 5)(r - 2)$$. - Rewrite equation: $$\frac{1}{r - 2} + \frac{1}{(r - 5)(r - 2)} = \frac{6}{r - 2}$$. - Multiply first term by $$\frac{r - 5}{r - 5}$$: $$\frac{r - 5}{(r - 2)(r - 5)} + \frac{1}{(r - 5)(r - 2)} = \frac{6}{r - 2}$$. - Combine left side: $$\frac{r - 5 + 1}{(r - 2)(r - 5)} = \frac{6}{r - 2}$$. - Simplify numerator: $$\frac{r - 4}{(r - 2)(r - 5)} = \frac{6}{r - 2}$$. - Multiply both sides by $$(r - 2)$$: $$\frac{r - 4}{r - 5} = 6$$. - Multiply both sides by $$(r - 5)$$: $$r - 4 = 6(r - 5)$$. - Expand right side: $$r - 4 = 6r - 30$$. - Solve for $$r$$: $$-4 + 30 = 6r - r \Rightarrow 26 = 5r \Rightarrow r = \frac{26}{5} = 5.2$$. Final answers: 1. $$v = 21$$ 2. $$m = 5$$ 3. $$n = 2$$ 4. $$r = \frac{26}{5}$$