1. **Problem 19:** Find the equation of combined variation where $y$ varies directly as $x$ and inversely as $z$, given $y=6$ when $x=8$ and $z=4$. Then find $y$ when $x=12$ and $z=9$. 2. **Problem 20:** Find the equation of combined variation where $y$ varies directly as $x$ and inversely as $z$, given $y=4$ when $x=3$ and $z=6$. Then find $y$ when $x=5$ and $z=20$. 3. **Problem 21:** Find the equation of combined variation where $a$ varies directly as $b$ and inversely as $c$, given $a=5$ when $b=2$ and $c=4$. Then find $a$ when $b=5$ and $c=25$. --- ### Step-by-step solution for each problem: ### Problem 19: 1. The combined variation formula is $y = k \frac{x}{z}$ where $k$ is the constant of variation. 2. Use given values to find $k$: $6 = k \frac{8}{4} \Rightarrow 6 = 2k \Rightarrow k = 3$. 3. The equation is $y = 3 \frac{x}{z}$. 4. Find $y$ when $x=12$ and $z=9$: $$y = 3 \frac{12}{9} = 3 \times \frac{4}{3} = 4$$ ### Problem 20: 1. Use the same formula $y = k \frac{x}{z}$. 2. Find $k$ using $y=4$, $x=3$, $z=6$: $$4 = k \frac{3}{6} = k \times \frac{1}{2} \Rightarrow k = 8$$ 3. Equation: $y = 8 \frac{x}{z}$. 4. Find $y$ when $x=5$, $z=20$: $$y = 8 \frac{5}{20} = 8 \times \frac{1}{4} = 2$$ ### Problem 21: 1. The formula is $a = k \frac{b}{c}$. 2. Find $k$ using $a=5$, $b=2$, $c=4$: $$5 = k \frac{2}{4} = k \times \frac{1}{2} \Rightarrow k = 10$$ 3. Equation: $a = 10 \frac{b}{c}$. 4. Find $a$ when $b=5$, $c=25$: $$a = 10 \frac{5}{25} = 10 \times \frac{1}{5} = 2$$ --- ### Final answers: - Problem 19: $y=4$ - Problem 20: $y=2$ - Problem 21: $a=2$