What This Problem Teaches

• Linear programming fundamentals: Setting up objective functions and constraint systems
• Feasible region analysis: Finding where all constraints are simultaneously satisfied
• Corner point theorem: Understanding why optimal solutions occur at vertices
• Resource allocation: Making optimal decisions with limited materials and capacity
• Real-world optimization: Translating business constraints into mathematical models

Visualizing the Problem

Before diving into the algebra, let's graph what we're working with. This helps us see the "feasible region" — the area where all constraints are satisfied simultaneously.

The yellow shaded region represents all valid combinations of earrings and necklaces that satisfy our constraints. Notice how the optimal solution (starred point) sits at a corner of this region — this isn't a coincidence!

Solution: Method 1 — The Corner Point Method

Linear programming problems always have their optimal solutions at corner points of the feasible region. We'll find all corner points, then test which gives the highest profit.

Step 1 — Define variables and constraints

Let x = number of pairs of earrings, y = number of necklaces.

Step 2 — Set up the objective function

The problem asks us to maximize profit but doesn't give specific profit values. We'll assume reasonable values: $15 profit per pair of earrings and $20 profit per necklace.

Step 3 — Find all corner points

Corner points occur where constraint lines intersect. Let's find each intersection:

Step 4 — Test each corner point

Calculate the profit P = 15x + 20y at each corner point:

Solution: Method 2 — Graphical Slope Analysis

Instead of testing every corner point, we can use the slope of the profit line to determine which corner will be optimal.

Step 1 — Analyze the profit function slope

Rewrite the profit function P = 15x + 20y in slope-intercept form:

The profit lines have a slope of -0.75. As we increase P, these lines shift upward and to the right.

Step 2 — Compare with constraint slopes

Find the slopes of our constraint boundaries:

Step 3 — Determine the optimal direction

Since the profit slope (-0.75) lies between the slopes of y = 20 (slope = 0) and the bead constraint (slope = -1.33), the optimal point must be at their intersection: (3, 20).

Verification

Let's verify our solution (3, 20) satisfies all constraints:

Resource usage check: Aubrey will use exactly 144 beads (24 for earrings + 120 for necklaces), make exactly 20 necklaces (her maximum), and only 3 pairs of earrings (well under her limit of 12).

Watch Out For These

The Math Beneath the Surface

This problem demonstrates the Fundamental Theorem of Linear Programming: If a linear program has an optimal solution, it occurs at a vertex (corner point) of the feasible region.

The slope analysis method works because the objective function creates a family of parallel lines. As you move these lines in the direction of optimization, the last point they touch in the feasible region is always a corner point.

How to Spot This Problem Type

Linear programming problems have distinctive markers that make them recognizable:

• "Maximize" or "minimize": Combined with constraints like "at most," "at least," or exact resource limits
• Resource allocation language: "Limited supply," "budget constraints," "capacity restrictions"
• Multiple decision variables: Usually asking how much of each type to make/buy/use
• Linear relationships: Each unit requires a fixed amount of resources (no economies of scale)
• Non-negativity implied: You can't make negative quantities in most real-world contexts

Variations you might see: profit maximization, cost minimization, diet problems (nutrition constraints), transportation problems (shipping costs), and production planning.

Real Applications

This exact mathematical structure appears throughout business and engineering:

• Manufacturing optimization: Car companies decide how many sedans vs. SUVs to produce given steel, labor, and factory constraints
• Portfolio management: Investment firms allocate money across asset classes subject to risk limits and regulatory requirements
• Supply chain logistics: Airlines determine flight schedules to maximize revenue while respecting aircraft availability and crew regulations
• Agriculture planning: Farmers choose which crops to plant based on land, water, and labor constraints to maximize profit

What If?