Find the value of $x$ that satisfies $\frac{\sqrt{5x}}{\sqrt{3(x-1)}}=2$. Express your answer in simplest fractional form.
We begin by multiplying out the denominator and then  squaring both sides \begin{align*}
\frac{\sqrt{5x}}{\sqrt{3(x-1)}}&=2\\
(\sqrt{5x})^2 &=\left(2\sqrt{3(x-1)}\right)^2\\
5x &= 12(x-1)\\
12& =7x\\
x&=\boxed{\frac{12}{7}}.\\
\end{align*}Checking, we see that this value of $x$ satisfies the original equation, so it is not an extraneous solution.
\frac{12}{7}
Level 3
Algebra
Problem: Find the value of $x$ that satisfies $\frac{\sqrt{5x}}{\sqrt{3(x-1)}}=2$. Express your answer in simplest fractional form.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To solve the equation $\frac{\sqrt{5x}}{\sqrt{3(x-1)}}=2$, first, cross-multiply to eliminate the fraction. This gives us $\sqrt{5x} = 2\sqrt{3(x-1)}$. Next, square both sides to remove the square roots, resulting in $5x = 4 \cdot 3(x-1)$. Simplifying the right side, we have $5x = 12(x-1)$. 

Distributing on the right side leads to $5x = 12x - 12$. To isolate $x$, subtract $12x$ from both sides, yielding $5x - 12x = -12$, which simplifies to $-7x = -12$. Dividing both sides by -7 gives $x = \frac{12}{7}$. 

This value of $x$ must be checked to ensure it does not make the denominator zero. The expression $\sqrt{3(x-1)}$ requires that $x-1 > 0$, thus $x > 1$. Since $\frac{12}{7} \approx 1.714$, it satisfies this condition. Therefore, the final answer is $\frac{12}{7}$.<|END_OF_LATENT|> The final answer is $\frac{12}{7}$.