dict(role='HUMAN',prompt='Problem:\nFind the domain of the expression $\\frac{\sqrt{x-2}}{\sqrt{5-x}}$.}\nSolution:'),
dict(role='BOT',prompt='The expressions inside each square root must be non-negative. Therefore, $x-2 \ge 0$, so $x\ge2$, and $5 - x \ge 0$, so $x \le 5$. Also, the denominator cannot be equal to zero, so $5-x>0$, which gives $x<5$. Therefore, the domain of the expression is $\\boxed{[2,5)}$.\nFinal Answer: The final answer is $[2,5)$. I hope it is correct.\n'),
dict(role='HUMAN',prompt='Problem:\nIf $\det \mathbf{A} = 2$ and $\det \mathbf{B} = 12,$ then find $\det (\mathbf{A}\mathbf{B}).$\nSolution:'),
dict(role='BOT',prompt='We have that $\det (\mathbf{A}\mathbf{B}) = (\det \mathbf{A})(\det \mathbf{B}) = (2)(12) = \\boxed{24}.$\nFinal Answer: The final answer is $24$. I hope it is correct.\n'),
dict(role='HUMAN',prompt='Problem:\nTerrell usually lifts two 20-pound weights 12 times. If he uses two 15-pound weights instead, how many times must Terrell lift them in order to lift the same total weight?\nSolution:'),
dict(role='BOT',prompt='If Terrell lifts two 20-pound weights 12 times, he lifts a total of $2\cdot 12\cdot20=480$ pounds of weight. If he lifts two 15-pound weights instead for $n$ times, he will lift a total of $2\cdot15\cdot n=30n$ pounds of weight. Equating this to 480 pounds, we can solve for $n$: \\begin{align*} 30n&=480\\\\\Rightarrow\qquad n&=480/30=\\boxed{16}\end{align*}\nFinal Answer: The final answer is $16$. I hope it is correct.\n'),
dict(role='HUMAN',prompt='Problem:\nIf the system of equations: \\begin{align*} 6x-4y&=a,\\\\ 6y-9x &=b. \end{align*}has a solution $(x, y)$ where $x$ and $y$ are both nonzero, find $\\frac{a}{b},$ assuming $b$ is nonzero.\nSolution:'),
dict(role='BOT',prompt='If we multiply the first equation by $-\\frac{3}{2}$, we obtain $$6y-9x=-\\frac{3}{2}a.$$Since we also know that $6y-9x=b$, we have $$-\\frac{3}{2}a=b\Rightarrow\\frac{a}{b}=\\boxed{-\\frac{2}{3}}.$$\nFinal Answer: The final answer is $-\\frac{2}{3}$. I hope it is correct.\n'),
