If the custody and child support arrangement was pursuant to a stipulation, private school or education costs should have been handled therein and you should bring that agreement to an attorney for review. If it was decision after trial, then this might not have been an issue that was brought up at the trial. We need more information.
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Unless your obligation to pay is tied to your consent, her violation of the joint custodial provisions are only relevant if you file a violation petition based on your custody order. As for the fact that you have 3 other people that you are supporting, that is also of no moment. You are not obligated to support your step child and presumably she has a dad who is paying support. The fact that you have another child does not change the support payable to your other child either. Also, the fact that you make more money does not, in and of itself, entitle her to an increase, unless your agreement or order state that it does. As for private school expenses, those expenses are not necessary and the mother would have to establish that the child has certain needs that only a private school can meet or that you consented to the private school. Otherwise, she will not prevail. Hire an attorney.
Yes, the judge will take it into consideration, however a lot of the judge's decision will depend on what you originally agreed to in the separation agreement. Further, some judge's might find it is in the best interest of your child to go to private school and force you to pay anyway.
You are not obligated to pay for private school if you never have before. The issue of upward modification is different, however, the Court should consider your other financial considerations. However, the first child is entitled to paramount consideration.