Make $x$ the subject of the formula where $x$ is positive:
$y=\sqrt{\frac{3 x-6}{4 x}}$
We are given the equation \(y = \sqrt{\frac{3x - 6}{4x}}\) and are tasked with making \(x\) the subject of the formula. We are given the condition that \(x\) is a positive real number. For the expression under the square root to be well-defined, the radicand \(\frac{3x - 6}{4x}\) must be non-negative. Given \(x > 0\), the denominator \(4x\) is always positive, so the numerator \(3x - 6\) must be greater than or equal to zero, which leads to \(3x \geq 6\), or \(x \geq 2\). This confirms the domain is valid for \(x > 0\). Our goal is to isolate \(x\) on one side of the equation.
Start with the original equation:
\[y = \sqrt{\frac{3x - 6}{4x}}\]
To eliminate the square root, we square both sides of the equation:
\[y^2 = \frac{3x - 6}{4x}\]
Next, we eliminate the fraction by multiplying both sides by the denominator, \(4x\):
\[4x \cdot y^2 = 3x - 6\]
\[4xy^2 = 3x - 6\]
To isolate the terms containing \(x\), we move all terms with \(x\) to one side of the equation. We can subtract \(4xy^2\) from both sides and add 6 to both sides, or rearrange as follows:
\[6 = 3x - 4xy^2\]
Now, factor out \(x\) on the right side:
\[6 = x(3 - 4y^2)\]
Finally, solve for \(x\) by dividing both sides by the expression \((3 - 4y^2)\):
\[x = \frac{6}{3 - 4y^2}\]
Thus, we have successfully made \(x\) the subject of the formula.
Rearranging formulas is a fundamental skill in science and engineering. For example, if you have a formula for the speed of a sound wave in a specific material based on its elasticity \(x\), you might need to solve for \(x\) to determine the material's properties from measured speed data \(y\). Being able to manipulate algebraic equations allows you to invert relationships, making it possible to find unknown parameters whenever you have measured experimental outcomes.