Table Seating: Solving a System of Equations

What This Looks Like

What You Will Learn

• How to translate a word problem into a system of linear equations
• Identifying count constraints versus value constraints in real-world scenarios
• Using the elimination method to solve systems efficiently
• Verifying solutions by substituting back into original equations
• Recognizing when a problem has exactly one solution and why

Solution: Method 1 — The Elimination Approach

Step 1 — Define variables and identify constraints

Let x = number of round tables and y = number of rectangular tables. We have two pieces of information to work with: the total number of tables (25) and the total number of seats (190).

Step 2 — Write the system of equations

The first constraint is about counting tables: x + y = 25

The second constraint is about counting seats: round tables contribute 6 seats each, rectangular tables contribute 8 seats each, for a total of 190 seats: 6x + 8y = 190

Step 3 — Use elimination to remove one variable

To eliminate x, multiply the first equation by -6:

Step 4 — Add the equations to eliminate x

Now add this modified equation to the second original equation:

Step 5 — Find the remaining variable

Substitute y = 20 back into the first equation:

Solution: Method 2 — Substitution Strategy

Step 1 — Solve the simpler equation for one variable

From the first equation x + y = 25, we can express x in terms of y:

Step 2 — Substitute into the second equation

Replace x in the second equation:

Step 3 — Find the first variable

Use y = 20 to find x:

Verification

Let's check our solution by substituting x = 5 and y = 20 back into both original equations:

Both equations are satisfied, confirming our solution is correct.

Does This Seem Reasonable?

Let's think about whether 5 round tables and 20 rectangular tables makes sense. The rectangular tables have more seats (8 vs 6), so if we want to maximize seating with a fixed number of tables, we'd expect more rectangular tables than round ones. Indeed, we found 20 rectangular tables versus only 5 round tables.

As a boundary check: if all 25 tables were round, we'd have 25 × 6 = 150 seats (too few). If all were rectangular, we'd have 25 × 8 = 200 seats (too many). Our answer of 190 seats falls between these extremes, which makes sense.

Common Pitfalls

The Pattern Behind This

This is a classic "mixture" or "constraint satisfaction" problem. The general pattern is:

This pattern appears in many contexts: coin problems (pennies and dimes), ticket sales (adult and child tickets), and resource allocation (workers with different productivity rates). The mathematical structure is always the same: two unknowns, two linear constraints, yielding exactly one solution at their intersection point.

Real Applications

Event Planning: Restaurants and venues use this calculation to optimize table arrangements for different party sizes and revenue targets.

Manufacturing: Production managers solve similar systems when allocating resources between different product lines with varying capacity requirements.

Transportation: Airlines and buses mix vehicle types (small and large capacity) to meet passenger demand while minimizing fleet size.

2026-08-25