in what sgnificant ways does the patriarchal system influence the Duchess's circumstances in the play?

The patriarchal system has great influence on the Duchess' life. First, she's a widow. She has nothing that she hasn't been left by her husband. She has no friends and no rights. She has been forbidden remarriage because her brothers can call that decision; they are her closest male relatives. Why have they forbidden her to remarry regardless that she is a young woman with small children? Well, why would they allow something that wouldn't be to their own benefit?

As a widow, the Duchess would for the first time have control over her own property. This would have been quite a change for someone always under the thumb of her father or husband. Widows were independent; they could make contracts; they could remarry and safeguard what the entered the marriage with. None-the-less, the patriarchal society that dominated the 1700's kept her from doing what she wanted initially. In the end she turned her nose up at her brothers' demands and married her steward. Independent? Yes, she was, she not only married against her brothers' wishes, but she married what they would consider beneath her as well. She will spend an inordinate amount of time trying to run her own life, but in the end her brothers undo what she has worked so hard for.