Certainly! Let's start by looking at the equation given:
[tex]\[
\left(p + \frac{a}{v^2}\right)(v - b) = \text{constant}
\][/tex]
We need to solve the equation for [tex]\( p \)[/tex].
### Step-by-Step Solution:
1. Expand the Given Equation:
First, let's expand the left side of the equation:
[tex]\[
\left(p + \frac{a}{v^2}\right)(v - b) = \text{constant}
\][/tex]
[tex]\[
p(v - b) + \frac{a}{v^2}(v - b) = \text{constant}
\][/tex]
2. Distribute Each Term:
Now distribute [tex]\( p \)[/tex] and [tex]\(\frac{a}{v^2}\)[/tex] across [tex]\( (v - b) \)[/tex]:
[tex]\[
p \cdot v - p \cdot b + \frac{a \cdot v}{v^2} - \frac{a \cdot b}{v^2} = \text{constant}
\][/tex]
Simplify the terms:
[tex]\[
p \cdot v - p \cdot b + \frac{a}{v} - \frac{a \cdot b}{v^2} = \text{constant}
\][/tex]
3. Isolate the [tex]\( p \)[/tex] Terms:
We need to isolate [tex]\( p \)[/tex] on one side of the equation. So let's collect all [tex]\( p \)[/tex]-related terms on one side:
[tex]\[
p(v - b) = \text{constant} - \frac{a}{v} + \frac{a \cdot b}{v^2}
\][/tex]
4. Solve for [tex]\( p \)[/tex]:
Divide both sides of the equation by [tex]\((v - b)\)[/tex] to solve for [tex]\( p \)[/tex]:
[tex]\[
p = \frac{\text{constant} - \frac{a}{v} + \frac{a \cdot b}{v^2}}{v - b}
\][/tex]
5. Combine the Terms:
Combine the terms in the numerator into a single fraction:
[tex]\[
p = \frac{\text{constant}v^2 - av + ab}{v^2(v - b)}
\][/tex]
Since we need to represent it cleanly, let’s reformat:
[tex]\[
p = \frac{-a \cdot b + a \cdot v - \text{constant} \cdot v^2}{v^2 (b - v)}
\][/tex]
Since dividing by [tex]\((b - v)\)[/tex] is the same as flipping the signs in the numerator, we have:
[tex]\[
p = \frac{-a \cdot b + a \cdot v - \text{constant} \cdot v^2}{v^2 (b - v)}
\][/tex]
This is a detailed step-by-step solution for isolating [tex]\( p \)[/tex] in the provided equation.
