Answer: Eradicating ivy (Hedera helix) is not an easy thing to do. It is deep rooted and will require extensive efforts to remove. You can try to dig it out, however it usually regrows from fragments of root left behind in the soil so it will be an ongoing process. You can treat it with an herbicide containing glyphosate, but it usually requires more than one treatment. Be sure to read the label directions and follow them carefully.
Keep in mind that ivy is usually planted in areas where grass just will not thrive --for example lawn grass grows poorly in shade, on steep slopes, in poor soil, under trees with surface roots, and so on. Ivy has been a traditional groundcover for areas like this, so it may be that your ivy was planted for this type of reason. Be sure to evaluate the site with regard to feasibility of growing lawn grass there before you rip it out.
In the meantime to control it encroaching, plan on a regular routine of cutting it back. The long runners can be easily pulled up while they are young.
I hope this helps with your planning.
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