Did you know?
When given a problem like finding two positive numbers such that their sum and twice the second number equals a constant (in this case, 64), and the goal is to maximize their product, the equation to optimize can be set up as follows:
Let's assume the two positive numbers are x and y. According to the problem statement, the first number added to twice the second number equals 64, so we can write the equation as x + 2y = 64.
To optimize their product, we need an equation to represent it. Since the product of two numbers can be expressed as xy, we now have the equation to optimize as xy.
By using algebraic techniques such as substitution or elimination, we can solve the system of equations consisting of x + 2y = 64 and xy, and find the values of x and y that maximize their product. This allows us to find the optimal solution and the corresponding numbers.
