Suppose the objective function P= xy is subject to the constraint 10x + y = 100, where x and y are real numbers.

a. Eliminate the variable y from the objective function so that P is expressed as a function of one variable x.

Verified step by step guidance

Start with the constraint equation: 10x + y = 100.

Solve the constraint equation for y in terms of x: y = 100 - 10x.

Substitute the expression for y from the constraint into the objective function P = xy.

This substitution gives P = x(100 - 10x).

Simplify the expression to express P as a function of x: P(x) = 100x - 10x^2.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Objective Function

An objective function is a mathematical expression that defines a quantity to be maximized or minimized, often subject to certain constraints. In this case, the objective function P = xy represents a product of two variables, x and y, which we aim to optimize under the given constraint.

Constraints

Constraints are conditions or limitations placed on the variables of an optimization problem. Here, the constraint 10x + y = 100 restricts the values that x and y can take, ensuring that any solution must satisfy this linear equation while optimizing the objective function.

Substitution Method

The substitution method involves replacing one variable in an equation with an expression derived from another equation. In this problem, we will use the constraint to express y in terms of x, allowing us to rewrite the objective function P solely in terms of x, simplifying the optimization process.