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Question 1209068: Solve 3a + 4b = 5a - 6b + 24 + a/(b^3 - 7b + 24) for a in terms of b.
Answer by yurtman(42)   (Show Source):
You can put this solution on YOUR website! To solve for `a`, we'll first group the terms involving `a` on one side of the equation:
```
3a + 4b = 5a - 6b + 24 + a/(b^3 - 7b + 24)
3a - 5a - a/(b^3 - 7b + 24) = 24 - 6b - 4b
```
Combine like terms:
```
-3a - a/(b^3 - 7b + 24) = 24 - 10b
```
Factor out `a` on the left side:
```
a(-3 - 1/(b^3 - 7b + 24)) = 24 - 10b
```
To isolate `a`, divide both sides by the factor `(-3 - 1/(b^3 - 7b + 24))`:
```
a = (24 - 10b) / (-3 - 1/(b^3 - 7b + 24))
```
To simplify the complex fraction, we can multiply the numerator and denominator by `(b^3 - 7b + 24)`:
```
a = [(24 - 10b)(b^3 - 7b + 24)] / [(-3)(b^3 - 7b + 24) - 1]
```
Expanding the numerator and simplifying the denominator:
```
a = (24b^3 - 168b + 576 - 10b^4 + 70b^2 - 240b) / (-3b^3 + 21b - 72 - 1)
```
Combining like terms:
```
a = (-10b^4 + 24b^3 + 70b^2 - 408b + 576) / (-3b^3 + 21b - 73)
```
So, the solution for `a` in terms of `b` is:
```
a = (-10b^4 + 24b^3 + 70b^2 - 408b + 576) / (-3b^3 + 21b - 73)
```
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