Precalculus Study Notes: Linear Systems, Word Problems, and Linear Programming

Linear Equations and Systems

Solving Systems of Linear Equations

Systems of linear equations are sets of two or more equations with multiple variables. Solving these systems allows us to find values for the variables that satisfy all equations simultaneously.

• Definition: A system of linear equations consists of two or more linear equations with the same set of variables.

• Methods of Solution:

  • Substitution: Solve one equation for one variable and substitute into the other.

  • Elimination: Add or subtract equations to eliminate a variable.

  • Graphical: Plot both equations and find the intersection point.

• Example: Find two numbers whose sum is 12 and difference is 4.

  • Let $x$ and $y$ be the numbers.

  • $x + y = 12$

  • $x - y = 4$

  • Solving: $x = 8$, $y = 4$

Word Problems Involving Linear Equations

Many real-world problems can be modeled using linear equations. These often involve costs, quantities, or measurements.

• Example: At a restaurant, the cost for a breakfast taco and a small glass of milk is $2.10. The cost for 2 tacos and 3 small glasses of milk is $5.15. Find the cost of a taco and a small glass of milk.

  • Let $x$ = cost of taco, $y$ = cost of milk.

  • $x + y = 2.10$

  • $2x + 3y = 5.15$

  • Solving: $x = 1.15$, $y = 0.95$

Applications of Linear Equations

Business and Inventory Problems

Linear equations are used to solve problems involving sales, inventory, and cost calculations.

• Example: The Frosty Ice-Cream Shop sells sundaes for $2 and banana splits for $3. On a hot summer day, the shop sold 8 more sundaes than banana splits and made $100.

  • Let $x$ = number of sundaes, $y$ = number of banana splits.

  • $x = y + 8$

  • $2x + 3y = 100$

  • Solving: $x = 36$, $y = 28$

Geometry and Measurement Problems

Linear equations can be used to solve geometric problems involving perimeter, area, and dimensions.

• Example: The perimeter of a rectangular wooden deck is 90 feet. The deck's length, $l$, is 5 feet less than 4 times its width, $w$. Write and solve a system of linear equations to determine the dimensions.

  • $2l + 2w = 90$

  • $l = 4w - 5$

  • Solving: $l = 55$ ft, $w = 10$ ft

Linear Programming

Introduction to Linear Programming

Linear programming is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships.

• Key Terms:

  • Constraints: Linear inequalities that restrict the values of variables.

  • Objective Function: The function to be maximized or minimized (e.g., profit, cost).

  • Feasible Region: The set of all possible solutions that satisfy the constraints.

• Steps in Linear Programming:

  1. Define variables.

  2. Write constraints as inequalities.

  3. Write the objective function.

  4. Graph the feasible region.

  5. Find the optimal value at a vertex of the feasible region.

Examples of Linear Programming Problems

• Example 1: Bob builds tool sheds. He has 10 sheets of dry wall and 15 studs for a small shed, and 15 sheets of dry wall and 45 studs for a large shed. He has available 60 sheets of dry wall and 135 studs. If he makes $390 profit on a small shed and $520 on a large shed, how many of each type should he build to maximize profit?

  • Let $x$ = number of small sheds, $y$ = number of large sheds.

  • Constraints:

    • $10x + 15y \leq 60$ (dry wall)

    • $15x + 45y \leq 135$ (studs)

  • Objective function: $P = 390x + 520y$

  • Solution: 3 small and 2 large sheds

• Example 2: A toy manufacturer wants to minimize cost for producing two lines of toy airplanes. Each Flying Falcon costs $600$ to build. What is the minimum cost per day?

  • Let $x$ = number of Flying Falcons, $y$ = number of Flying Hawks.

  • Constraints based on production limits.

  • Objective function: $C = 600x$

  • Solution: $600$

Tables: Linear Programming Solutions

Summary Table of Linear Programming Problems

Problem	Variables	Constraints	Objective Function	Optimal Solution
Tool Sheds	Small sheds ($x$), Large sheds ($y$)	$10x + 15y \leq 60$ $15x + 45y \leq 135$	$390x + 520y$	3 small, 2 large
Toy Wagons	Small wagons ($x$), Large wagons ($y$)	$4x + 6y \leq 60$ $x \geq 6$ $y \geq 4$	$12x + 30y$	6 small, 4 large
Farm Planting	Acres of wheat ($x$), Acres of rye ($y$)	$x + y \leq 120$ $x \geq 0$ $y \geq 0$	$357.53x + 139.32y$	40 acres wheat, 4 acres rye
Ore Processing	Tons from source A ($x$), Tons from source B ($y$)	$x + y \leq 6$ $0.2x + 0.5y \geq 2$	$200x + 300y$	2 tons A, 4 tons B
Factory Products	Products from Factory X ($x$), Products from Factory Y ($y$)	$x + y \leq 1200$ $x \geq 0$ $y \geq 0$	$2x + 3y$	1200 products from Factory X, 0 from Factory Y

Additional Info

• Linear programming problems often require graphing the feasible region and checking the vertices for optimal solutions.

• Constraints can be equalities or inequalities, and the solution must satisfy all constraints.

• Objective functions are typically maximized (profit) or minimized (cost).