Funny you used the word eradicate...DDT will eradicate the aphids everything else just controls them ;-).
Orthene may last a couple of days. No it isn't systemic throughout the plant. Not it isn't a good idea to try and kill something that isn't there.
I'd be care not to fertilize your plants with a high nitrogen fertilizer because all the tender growth attracks the bugs. Sometimes even with our best efforts we get aphids (and other bugs) anyway.

You can always use horticultural soap-little guns
Or Merit/Imidacoprid as a spray or as a ground drench (which is systemic through the whole plant)-big guns.

Aphids are soft bodied insects and die when their bodies come in contact with soap . It clogs their ability to breathe I was told .
In a hose end sprayer add the cheapest dish soap you can find - a few squirts will do . Use the kind made for washing dishes in the sink . The sprayer will generate lots of suds . Cover the affected areas with suds - i.e. the growth parts of fresh green leaves . This will only kill the current aphids and not prevent new infestation .
It is safe to spray as often as you need . Spray after it rains also . Once the new growth becomes mature the aphids should go away .

Well, it says it is systemic on the package of Orthene. I have these beautiful crape myrtles-they have grown to 25 feet tall in just 3 or 4 years, but they always flower so poorly! Every year they look terrible because they get covered in that black film from the aphid honeydew, and all the shrubs below it get the black crap on them too. They are Muskogee, and although they do get a little shade from some pines in the next yard, they still get a lot of sun. It's hard for trees that tall not to get sun! I know a lack of sun would be most people's first thought as to why the flowering is poor.

Is it possible the quick growth rate is affecting the flowering? This year they do seem to have slowed down some. Maybe as they are reaching maturity they will flower better. I of course hate 'crape murder' but I do like to cut back some of the long, thin branches. Unfortunately the trees have gotten so tall I can't reach the one at the top of the tree. I'd like to take a couple of feet off the top.

Please--give me some advice on taking care of giant crape myrtles!

I'm not crepe myrtle expert but you can prune it w/o murdering it. I know what you mean about getting so tall. I had one I just gave up on pruning at all.
Maybe you need a fertilizer with a high middle number (although those are becoming a no-no because of leaching!)
Systemic-it is absorbed into the plant leaves...it doesn't just sit on the leaf. There are some systemics that do go throughout the whole plant they are generally applied as a soil/root drench.
Sorry about your CM problems...maybe they will start blooming more w. time.

Hi. I discovered lots of aphids on my artichokes and periwinkles a few days ago. I sprayed them not only with plain dish soap and water, but with 1 quart water, 1 tsp mild dish soap, and 1/4 cup vegetable oil (such as canola). This solution seems to coat the leaves better with more viscosity and destroys the aphids faster. I guess you know it's really important to thoroughly spray the undersides of the leaves. Take care, Sheri :-)