Make $x$ the subject of the formula where $x$ is positive:
$y=\sqrt{\frac{3 x-6}{4 x}}$
We are given the equation \(y = \sqrt{\frac{3x - 6}{4x}}\) where \(x\) is a positive real number (\(x > 0\)). Our goal is to isolate \(x\) to make it the subject of the formula. The expression inside the square root, \(\frac{3x - 6}{4x}\), must be non-negative for \(y\) to be a real number, which implies \(3x - 6 \ge 0\) because \(x > 0\) ensures the denominator \(4x\) is positive. Therefore, \(3x \ge 6\), which simplifies to \(x \ge 2\).
To isolate \(x\), we first eliminate the square root by squaring both sides of the equation:
\[y^2 = \frac{3x - 6}{4x}\]
Next, we clear the fraction by multiplying both sides by \(4x\):
\[4xy^2 = 3x - 6\]
Now, we collect all terms involving \(x\) on one side of the equation. We subtract \(3x\) from both sides:
\[4xy^2 - 3x = -6\]
Alternatively, adding \(6\) and subtracting \(4xy^2\) from both sides gives:
\[3x - 4xy^2 = 6\]
We can now factor out \(x\) from the left side:
\[x(3 - 4y^2) = 6\]
Finally, we divide both sides by the expression \((3 - 4y^2)\) to solve for \(x\):
\[x = \frac{6}{3 - 4y^2}\]
Since the original equation involves a square root that must return a non-negative value (\(y \ge 0\)), and we know \(x \ge 2\) from our initial analysis, the expression \(x = \frac{6}{3 - 4y^2}\) correctly represents the relationship between \(x\) and \(y\). Thus, the subject of the formula is \(x = \frac{6}{3 - 4y^2}\).
This type of algebraic manipulation is crucial in physics and engineering when rearranging formulas to solve for a specific variable. For example, if you have a formula for a physical quantity (like velocity or acceleration) involving a square root, such as \(v = \sqrt{\frac{2F}{m}}\), you can use these exact steps to solve for mass \(m\) when you know the velocity and force. Mastering this skill allows for greater flexibility when analyzing data and applying theoretical models to real-world problems.