To isolate \( f \) in the equation \(\frac{1}{u} = \frac{1}{f} - \frac{1}{v}\), we will follow a series of algebraic steps.
1. Initial Equation:
[tex]\[
\frac{1}{u} = \frac{1}{f} - \frac{1}{v}
\][/tex]
2. Combine the right-hand side over a common denominator:
To combine the terms \(\frac{1}{f}\) and \(\frac{1}{v}\), we need a common denominator:
[tex]\[
\frac{1}{f} - \frac{1}{v} = \frac{v - f}{fv}
\][/tex]
Therefore, the equation becomes:
[tex]\[
\frac{1}{u} = \frac{v - f}{fv}
\][/tex]
3. Equate the two fractions:
Since both sides are fractions, we can equate the numerators:
[tex]\[
\frac{1}{u} = \frac{v - f}{fv}
\][/tex]
4. Cross-multiply to clear the fractions:
To clear the fractions, multiply both sides by \( u \cdot fv \):
[tex]\[
fv = u(v - f)
\][/tex]
5. Distribute \( u \) on the right-hand side:
[tex]\[
fv = uv - uf
\][/tex]
6. Group terms involving \( f \) on one side:
Add \( uf \) to both sides to isolate terms involving \( f \):
[tex]\[
fv + uf = uv
\][/tex]
Factor out \( f \) on the left-hand side:
[tex]\[
f(v + u) = uv
\][/tex]
7. Solve for \( f \):
Divide both sides by \( v + u \):
[tex]\[
f = \frac{uv}{u + v}
\][/tex]
Thus, the subject \( f \) is:
[tex]\[
f = \frac{uv}{u + v}
\][/tex]
