Make $x$ the subject of the formula where $x$ is positive:
$y=\sqrt{\frac{3 x-6}{4 x}}$
The goal is to express the variable \(x\) in terms of \(y\) starting from the equation \(y = \sqrt{\frac{3x - 6}{4x}}\), given that \(x > 0\). First, note that since the expression is a square root, we must have \(\frac{3x - 6}{4x} \geq 0\). Because \(x > 0\), the numerator must be non-negative, meaning \(3x - 6 \geq 0\), which simplifies to \(x \geq 2\). This also implies that \(y \geq 0\).
To isolate \(x\), we first square both sides of the equation to eliminate the radical:
\[y^2 = \frac{3x - 6}{4x}\]
Next, we clear the fraction by multiplying both sides by the denominator \(4x\):
\[4xy^2 = 3x - 6\]
Now, we bring all terms involving \(x\) to the left side of the equation:
\[4xy^2 - 3x = -6\]
To isolate \(x\), we factor it out from the left side:
\[x(4y^2 - 3) = -6\]
Finally, we divide both sides by \((4y^2 - 3)\) to solve for \(x\):
\[x = \frac{-6}{4y^2 - 3}\]
To simplify this expression, we multiply the numerator and the denominator by \(-1\):
\[x = \frac{6}{3 - 4y^2}\]
Thus, \(x\) expressed in terms of \(y\) is \(x = \frac{6}{3 - 4y^2}\).
This type of formula manipulation is very common in physics and engineering, particularly when you need to rearrange a relationship between physical quantities to find a specific parameter. For example, if you have a formula for the speed of an object in a constrained system (like a pendulum or a rolling ball) and you need to determine the required position or length of the system to achieve a specific speed, you would perform exactly this kind of algebraic rearrangement. Mastery of these steps allows you to easily solve for unknown variables in complex equations.