Sure, let's solve the problem step-by-step. We need to express \(\frac{3u - 2}{4u + 1}\) in terms of \(v\), given that \(u = \frac{31}{2v^{-1}}\).
### Step 1: Simplify the Expression for \(u\)
First, we start with the given expression for \(u\):
[tex]\[ u = \frac{31}{2v^{-1}} \][/tex]
Recall that \(v^{-1}\) is the same as \(\frac{1}{v}\), so we can rewrite the expression as:
[tex]\[ u = \frac{31}{2 \cdot \frac{1}{v}} = \frac{31}{\frac{2}{v}} \][/tex]
When dividing by a fraction, we multiply by its reciprocal:
[tex]\[ u = 31 \cdot \frac{v}{2} = \frac{31v}{2} \][/tex]
### Step 2: Substitute \(u\) in the Given Expression
Now we need to substitute \(u = \frac{31v}{2}\) into the expression \(\frac{3u - 2}{4u + 1}\).
First, substitute \(u\) in the numerator:
[tex]\[ 3u - 2 = 3\left(\frac{31v}{2}\right) - 2 = \frac{93v}{2} - 2 \][/tex]
Convert \(2\) to a fraction with a denominator of 2:
[tex]\[ 3u - 2 = \frac{93v}{2} - \frac{4}{2} = \frac{93v - 4}{2} \][/tex]
Next, substitute \(u\) in the denominator:
[tex]\[ 4u + 1 = 4\left(\frac{31v}{2}\right) + 1 = 2 \cdot 31v + 1 = 62v + 1 \][/tex]
### Step 3: Form the Final Expression
Now we can write the full expression by using the results from the previous steps:
[tex]\[ \frac{3u - 2}{4u + 1} = \frac{\frac{93v - 4}{2}}{62v + 1} \][/tex]
To simplify the fraction, multiply the numerator and the denominator by 2 to eliminate the complex fraction:
[tex]\[ \frac{\frac{93v - 4}{2}}{62v + 1} = \frac{93v - 4}{2(62v + 1)} \][/tex]
### Final Answer
Thus, \(\frac{3u - 2}{4u + 1}\) in terms of \(v\) is:
[tex]\[ \frac{93v - 4}{2(62v + 1)} \][/tex]
