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Linear Programming

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Linear Programming

Study Guide

Key Definition

A linear programming problem involves optimizing a linear function, called the objective function, subject to a system of linear constraints.

Important Notes

The objective function is of the form $f(x, y) = ax + by + c$
Feasible solutions are the set of $(x, y)$ that satisfy all constraints
Optimal values occur at the vertices of the feasible region
A bounded region is contained in a finite area, while an unbounded region extends infinitely
Evaluate the objective function at each vertex to find the minimum or maximum value

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

$\leq$ means less than or equal to

$\geq$ means greater than or equal to

Remember to use proper notation when solving problems

Why It Works

Linear programming helps find optimal solutions by evaluating the objective function at vertices of the feasible region.

Remember

The optimal solution occurs at a vertex of the feasible region.

Quick Reference

Objective Function:$f(x, y) = ax + by + c$

Understanding Linear Programming

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Video explanation of this concept

Beginner

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Beginner Explanation

Linear programming involves maximizing or minimizing a linear function subject to constraints.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the form of a linear objective function in linear programming?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A teenager wants to maximize their weekly allowance by doing chores. They can do $x$ hours of lawn mowing and $y$ hours of babysitting. Each hour of mowing earns $5 and each hour of babysitting earns $10. They cannot work more than 10 hours per week and cannot work negative hours. What is the maximum allowance the teenager can earn?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider a region defined by $x \geq 0$, $y \geq 0$, and $x + y \leq 10$. What is the maximum value of 3x + 2y?

Challenge Quiz

Single Choice Quiz

Advanced

Which of the following represents a constraint in linear programming?

Recap

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Review key concepts and takeaways