Got a question with substitution?

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The discussion revolves around using the Substitution method to maximize the expression 4x - 2xy + 3y under the constraint 4x - y = 2. The initial steps involve substituting y with 4x - 2 into the original equation. Participants suggest differentiating the resulting expression with respect to x to find critical points for maximum or minimum values. However, there is a debate on whether this problem is calculus-based or not, with some advocating for the use of the Lagrange method as a more elegant solution. Ultimately, the conversation highlights different approaches to solving optimization problems in mathematics.
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Using the Substitution method, find the maximum value of

4x-2xy+3y subject to the constraint 4x-y=2

I can do the 1st part:

4x-y=2--->
4x-2=y

substitute this into the original equation:
4x-2x(4x-2)+3(4x-2)

Hope that right so far! but don't know where to go from there? anyone could help please?

thanks ;)
 
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then you differentiate wrt x and equal to zero to find max/min
 
what do u mean by wrt?
 
"wrt" : "with respect to"
CartoonKid was assuming this was a calculus problem. The standard method for solving "maximum- minimum" problems in calculus is to differentiate and set the derivative equal to 0. But since you apparently weren't aware of that, I suspect this is not a calculus problem!

If you multiply y= 4x-2x(4x-2)+3(4x-2) out, you get a quadratic. If you are doing problems like this, I presume you know how to complete the square to find the vertex of the graph which, in this problem, gives you the maximum value.

So: multiply this out, then complete the square.
 
This method is totally counterproductive...The method discovered by Lagrange is much more elegant...

Daniel.
 
But one would need a rather intermediate mathematical background to use them...
 
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