Reducing variable to 1 term: simplifying a fraction
I have an equation P(x) that resembles:
(C1e-ax + C2e-bx)
(e-ax + e-bx)
where the horizontal line represents a division line. I do not know the values of a, b or C1 nor C2, but they all have the same units.
I am supposed to graph P as a function of x, for which I assume I need to collect all of the x terms into one x term.
Does anyone know what technique I could use to accomplish this or at least a first step in the right direction? I'd really appreciate it!
Thank you for your time and help.
Re: Reducing variable to 1 term: simplifying a fraction
I should mention that I looked into partial fraction decomposition, but it seemed to be the wrong direction as it doesn't solve my issue with multiple x-terms in the denominator.
I also tried separating the numerator, then factoring out the corresponding exponent in the denominator. This results in two fractions with only one x term in each denominator, but I am unsure how to proceed from there to reduce it further to only one term.
I think if I could get my feet grounded on a correct approach, I could figure it out. Thanks for reading.
Edit: I have solved the problem. Is it possible to write math equations into the forums? If so, I will show mathematically how I did it in case anyone is interested.
In words, I made the following steps:
1. factored e-bx into (e-ax)(e-cx). This allowed me to factor out e-ax in both the numerator and denominator and reduced me to two exponential terms.
2. I then factored out C1 from the numerator.
3. I simultaneously added and subtracted a C1 (C3e-cx) term in the numerator, where I chose C3 so that the numerator could be separated to cancel with the denominator. This left me with a non-fraction term and a simplified fraction term with no addition operators in the numerator.
4. Taking the simplified fraction, I factored out e-cx in the numerator and denominator and canceled it out.
This left me with an equation with one x-term as a reciprocal in the denominator. It looks like:
C1 + C3 / (1 + 1 / e-cx)