Alright, let's tackle each part step-by-step to solve for [tex]\( u \)[/tex].
### Part (a)
Given the equation:
[tex]\[ \frac{1}{t} = \frac{1}{u} + \frac{1}{y} \][/tex]
1. Start by isolating [tex]\(\frac{1}{u}\)[/tex]:
[tex]\[ \frac{1}{u} = \frac{1}{t} - \frac{1}{y} \][/tex]
2. To combine the fractions on the right-hand side, use a common denominator:
[tex]\[ \frac{1}{u} = \frac{y - t}{ty} \][/tex]
3. Now, take the reciprocal of both sides to solve for [tex]\( u \)[/tex]:
[tex]\[ u = \frac{ty}{y - t} \][/tex]
Since [tex]\(\frac{1}{u} = \frac{1}{t} - \frac{1}{y}\)[/tex], you can rewrite the expression more generally as:
[tex]\[ u = \frac{1}{\frac{1}{t} - \frac{1}{y}} \][/tex]
So the solution for part (a) is:
[tex]\[ u = \frac{1}{\frac{1}{t} - \frac{1}{y}} \][/tex]
### Part (b)
Given the equation:
[tex]\[ s = \frac{h}{2(u + v)} \][/tex]
1. Multiply both sides by [tex]\(2(u + v)\)[/tex] to remove the denominator:
[tex]\[ 2s(u + v) = h \][/tex]
2. Divide both sides by [tex]\(2s\)[/tex] to isolate [tex]\(u + v\)[/tex]:
[tex]\[ u + v = \frac{h}{2s} \][/tex]
3. Finally, solve for [tex]\( u \)[/tex] by subtracting [tex]\( v \)[/tex] from both sides:
[tex]\[ u = \frac{h}{2s} - v \][/tex]
So the solution for part (b) is:
[tex]\[ u = \frac{h}{2s} - v \][/tex]
These steps should clearly show how to solve for [tex]\( u \)[/tex] in both given equations.
