Your the parent, they are the children. You can explain until you are blue in the face, but most kids do not have the maturity or wisdom to be able to understand the reasoning behind all your decisions. But it doesn't matter. As the parent, you have to make the decisions that are right for your family. If they get it, great. If not, too bad. That's just one of the things in life that suck, and they are just going to have to learn to except it. If you want them to have a better understanding of "why," try doing the following for a month.
1. Start by waking them up at 5 a.m. every morning, including weekends. Make them get up, get dressed, and take them for a walk around the neighborhood for 15-30 minutes. Every morning, without fail.
2. Take away some of their privileges they very much enjoy that cost money, such as internet, cell phones, etc.
3. Do not allow them to go places away from the home to do things they enjoy such as hobbies or extracurricular activities.
4. When they are home, they are to take a walk around the neighborhood every few hours.
5. Make them vacuum one room each every single day.
Do this to give them just a tiny idea of just how much work and responsibility goes into having a dog. 1. Explain to them that the dog has to be walked every morning at the same time, including weekends. Just because it is the weekend, this doesn't mean the dog magically doesn't need to pee or poop anymore. 2. Explain to them that because it costs money to feed a dog, provide veterinary care, and hire a dog sitter/walker while everyone is at school and work, you are going to have to cut back on other expenses. 3. Explain to them that they can no longer just go wherever they want whenever they want, because they have a dog at home that needs to be taken care of, and cannot be left alone for long periods of time. 4. The dog will periodically have to go to the bathroom, and it doesn't matter what they have going on, they will have to stop and take the dog for a walk. 5. Dogs shed. In order to keep your house from becoming a big giant hairball and stinking like nasty dog all the time, you have to vacuum or sweep every single day.
After a month of getting up and going for a walk at 5 a.m., not having the same fun privileges, and having to clean every day, I'm sure they will rethink the idea of having a dog.