Reducing variable to 1 term: simplifying a fraction

Hi there,

I have an equation P(x) that resembles:

(C_{1}e^{-ax} + C_{2}e^{-bx})

(e^{-ax} + e^{-bx})

where the horizontal line represents a division line. I do not know the values of a, b or C_{1} nor C_{2}, 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 C_{1} from the numerator.

3. I simultaneously added and subtracted a C_{1} (C_{3}e^{-cx}) term in the numerator, where I chose C_{3} 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:

C_{1} + C_{3} / (1 + 1 / e^{-cx})