To make [tex]\( P \)[/tex] the subject of the formula [tex]\( L = \frac{2}{3} \sqrt{\frac{x^2 - P T}{y}} \)[/tex], follow these detailed steps:
1. Eliminate the fraction: Multiply both sides of the equation by [tex]\(\frac{3}{2}\)[/tex] to eliminate the fraction on the right-hand side.
[tex]\[
\left(\frac{3}{2}\right) L = \sqrt{\frac{x^2 - P T}{y}}
\][/tex]
2. Eliminate the square root: Square both sides of the equation to eliminate the square root.
[tex]\[
\left(\frac{3}{2} L\right)^2 = \frac{x^2 - P T}{y}
\][/tex]
Simplify the left-hand side:
[tex]\[
\left(\frac{3}{2} L\right)^2 = \frac{9}{4} L^2
\][/tex]
So the equation becomes:
[tex]\[
\frac{9}{4} L^2 = \frac{x^2 - P T}{y}
\][/tex]
3. Isolate [tex]\( x^2 - P T \)[/tex]: Multiply both sides by [tex]\( y \)[/tex] to isolate the term [tex]\( x^2 - P T \)[/tex] on the right-hand side.
[tex]\[
y \cdot \frac{9}{4} L^2 = x^2 - P T
\][/tex]
Simplify the left-hand side:
[tex]\[
\frac{9 y}{4} L^2 = x^2 - P T
\][/tex]
4. Rearrange to solve for [tex]\( P \)[/tex]: Isolate [tex]\( P T \)[/tex] on one side of the equation and solve for [tex]\( P \)[/tex].
[tex]\[
P T = x^2 - \frac{9 y}{4} L^2
\][/tex]
Now, divide both sides by [tex]\( T \)[/tex]:
[tex]\[
P = \frac{x^2 - \frac{9 y}{4} L^2}{T}
\][/tex]
Simplify the expression:
[tex]\[
P = \frac{x^2 - \left(\frac{9}{4} y L^2\right)}{T}
\][/tex]
Thus, the formula for [tex]\( P \)[/tex] as the subject is:
[tex]\[
P = \frac{x^2 - \left(\frac{9}{4} y L^2\right)}{T}
\][/tex]
