Begin by bringing a large pot of salted water to a boil. Add the halved or quartered potatoes and cook for about 10 minutes, or until tender when pierced with a fork.
While the potatoes are cooking, trim the ends off the green beans. After the potatoes have cooked for 10 minutes, add the green beans to the pot.
Continue to cook both potatoes and green beans for an additional 5-7 minutes, until the green beans are bright green and tender-crisp.
Drain the potatoes and green beans in a colander and set aside.
In the same pot, heat the olive oil over medium heat. Add the minced garlic and sauté for about 1 minute, just until fragrant but not browned.
Return the drained potatoes and green beans to the pot. Stir well to coat with the garlic oil. Season with salt and pepper, and add lemon juice if using.
Toss everything together gently, ensuring the vegetables are well combined with the seasonings. Cook for an additional 2 minutes to heat through.
Serve warm as a delicious side dish.