Undergrad Values of Lagrange multipliers when adding new constraints

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When adding a new constraint to a Lagrange function, the value of the existing Lagrange multiplier can change. The introduction of an additional constraint alters the optimization landscape, potentially affecting the optimal solution. Each constraint influences the multipliers associated with the objective function. Therefore, the original multiplier's value is not guaranteed to remain constant after modifying the constraints. This highlights the interdependence of constraints in optimization problems.
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Say we have a Lagrange function with one multiplier a times a constrain. I minimize and solve the system to find a. I now add another constrain to the same system multiplied by the constant b. Is the value of a the same or can it change?
 
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No it is not the same.
 

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