A farmer has 1200 acres of land on which he grows corn,...

A farmer has 1200 acres of land on which he grows corn, wheat, and soybeans. It costs $\$ 45$ per acre to grow corn, $\$ 60$ to grow wheat, and $\$ 50$ to grow soybeans. Because of market demand, the farmer will grow twice as many acres of wheat as of corn. He has allocated $\$ 63,750$ for the cost of growing his crops. How many acres of each crop should he plant?

Key Concepts

Budget Constraint

A budget constraint is a type of limitation that expresses the total available resources, such as money, which must cover the costs of various items or activities. In many allocation problems, the total expenditure is represented as a linear combination of the individual costs, forming an equation that the solution must satisfy.

Ratio Constraint

A ratio constraint imposes a fixed proportional relationship between two or more quantities. This type of constraint is common in resource allocation and planning problems where one quantity is required to be a specific multiple of another, and it is expressed algebraically as part of the system of equations.

System of Linear Equations

This concept involves setting up and solving multiple equations that express different relationships among the same set of unknowns. In many practical problems, including allocation or budgeting issues, various constraints are modeled as linear equations that must be satisfied simultaneously to find the solution.

Constraint Formulation from Word Problems

In applied mathematics and algebra, translating the verbal description of a problem into mathematical equations is crucial. This process involves identifying the relevant quantities and their relationships—such as total amounts, cost limitations, or fixed ratios—and then expressing them as constraints within a system of equations.

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