Certainly! Let's solve the equation [tex]\(\frac{1}{u} + \frac{1}{v} = \frac{1}{f}\)[/tex] for [tex]\(u\)[/tex] step-by-step.
1. Start with the given equation:
[tex]\[
\frac{1}{u} + \frac{1}{v} = \frac{1}{f}
\][/tex]
2. Subtract [tex]\(\frac{1}{v}\)[/tex] from both sides of the equation to isolate [tex]\(\frac{1}{u}\)[/tex]:
[tex]\[
\frac{1}{u} = \frac{1}{f} - \frac{1}{v}
\][/tex]
3. Find a common denominator for the right-hand side of the equation:
The common denominator for [tex]\(f\)[/tex] and [tex]\(v\)[/tex] is [tex]\(fv\)[/tex]. So, we can rewrite the right-hand side:
[tex]\[
\frac{1}{f} = \frac{v}{fv}, \quad \frac{1}{v} = \frac{f}{fv}
\][/tex]
Therefore,
[tex]\[
\frac{1}{u} = \frac{v}{fv} - \frac{f}{fv} = \frac{v - f}{fv}
\][/tex]
4. Simplify the right-hand side:
[tex]\[
\frac{1}{u} = \frac{v - f}{fv}
\][/tex]
5. Take the reciprocal of both sides to solve for [tex]\(u\)[/tex]:
Recall that [tex]\(\frac{1}{\left(\frac{a}{b}\right)} = \frac{b}{a}\)[/tex]. Applying this to our equation, we get:
[tex]\[
u = \frac{fv}{v - f}
\][/tex]
So, we have successfully solved for [tex]\(u\)[/tex] in terms of [tex]\(f\)[/tex] and [tex]\(v\)[/tex]:
[tex]\[
u = \frac{f \cdot v}{v - f}
\][/tex]
This is the formula for [tex]\(u\)[/tex].
