Lunch Order Problem: Solving a System of Linear Equations

What This Problem Teaches

• System of equations setup: Translating real-world constraints into mathematical equations
• Linear algebra fundamentals: Understanding how two independent constraints create exactly one solution
• Elimination method: Strategic manipulation to remove variables systematically
• Solution uniqueness: Recognizing when a system has exactly one answer versus infinitely many
• Real-world validation: Checking that mathematical answers make practical sense

Visualizing the Constraints

This table shows our constraints clearly: we need exactly 23 items total, and they must cost exactly $32.10. With two constraints and two unknowns, we have a system that should yield exactly one solution.

Solution: Method 1 — The Elimination Approach

Step 1 — Set up the system of equations

Let's define our variables clearly: b = number of hamburgers, h = number of hot dogs. Our constraints give us two equations:

Step 2 — Eliminate one variable using multiplication

To eliminate h, I'll multiply equation (1) by 1.10 so that both equations have 1.10h:

Step 3 — Subtract to eliminate h

Now I'll subtract equation (1a) from equation (2):

Step 4 — Solve for b

Dividing both sides by 0.40:

Step 5 — Find h using substitution

Substituting b = 17 back into equation (1):

Solution: Method 2 — The Substitution Strategy

Step 1 — Express one variable in terms of the other

From equation (1), b + h = 23, I can solve for h:

Step 2 — Substitute into the cost equation

Replace h in equation (2) with (23 - b):

Step 3 — Distribute and simplify

Step 4 — Solve for both variables

Both methods arrive at the same answer, confirming our solution.

Verification

Let's check both constraints with our answer: 17 hamburgers and 6 hot dogs.

Check the total count

Check the total cost

Both constraints are satisfied perfectly. The mathematical solution matches the real-world requirements.

Is This the Only Solution?

Yes, this combination is unique. Here's why:

To see this graphically, imagine plotting all possible combinations:

• The line b + h = 23 contains all combinations that feed exactly 23 people
• The line 1.50b + 1.10h = 32.10 contains all combinations that cost exactly $32.10
• Only one point lies on both lines simultaneously

Common Pitfalls

Mistake 1: Setting up equations incorrectly

Mistake 2: Decimal arithmetic errors

Mistake 3: Forgetting to answer the uniqueness question

The Math Beneath the Surface

This problem demonstrates a fundamental principle: n independent linear equations in n unknowns typically have exactly one solution.

In our problem:

• Equation 1: 1·b + 1·h = 23 (coefficients: 1, 1)
• Equation 2: 1.50·b + 1.10·h = 32.10 (coefficients: 1.50, 1.10)
• Determinant: (1)(1.10) - (1)(1.50) = -0.40 ≠ 0 ✓

Since the determinant is non-zero, we're guaranteed exactly one solution. This mathematical check confirms what we found algebraically.

Why This Matters Beyond School

This type of constraint satisfaction appears constantly in real situations:

• Resource allocation: Hospitals determining staffing levels with budget and patient care constraints
• Manufacturing: Factories optimizing production schedules with material and time limitations
• Project management: Balancing team size, budget, and timeline requirements
• Nutrition planning: Dietitians creating meal plans that meet caloric and nutritional targets

The systematic approach you've learned—identify constraints, set up equations, solve, and verify—applies directly to these professional contexts.

What-If Problems