Theater Ticket Sales: Solving a System of Equations

What This Problem Teaches

• Variable identification — recognizing the two unknown quantities and assigning meaningful variables
• Systems setup — translating word problems into two equations with two unknowns
• Substitution method — solving by isolating one variable and substituting
• Revenue calculations — understanding how unit price × quantity = total revenue
• Solution verification — checking answers against both original constraints

Visualizing the Setup

This table helps us see the structure clearly: we need equations for both the total ticket count and total revenue.

Solution: Method 1 — The Substitution Approach

Part A — Variable Representation

We need to define variables for the two unknown quantities:

Part B — Writing the System of Equations

We create two equations based on the given constraints:

Equation 1 comes from: adult tickets + child tickets = total tickets

Equation 2 comes from: (price per adult)(number of adults) + (price per child)(number of children) = total revenue

Part C — Finding the Solution

Step 1 — Isolate y from the first equation

From x + y = 400, we solve for y:

Step 2 — Substitute into the second equation

Replace y in the revenue equation with (400 - x):

Step 3 — Expand and simplify

Distribute the 5 and combine like terms:

Step 4 — Find y using substitution

Substitute x = 275 back into y = 400 - x:

Solution: Method 2 — The Elimination Approach

Starting with the same system, we can eliminate one variable by scaling equations:

Step 1 — Eliminate x by scaling equation (1)

Multiply equation (1) by -9 to make the x-coefficients opposites:

Step 2 — Add the equations

When we add, the x terms cancel out:

Step 3 — Find x using back-substitution

Substitute y = 125 into equation (1):

Both methods give us the same answer: 275 adult tickets and 125 child tickets.

Verification

Let's check our answer against both original constraints:

Both equations are satisfied, confirming our solution is correct.

Common Pitfalls

The Pattern Behind This

All two-category mixture problems follow the same structure:

This pattern applies whether you're mixing tickets, coins, investments, or chemical solutions. The key insight is recognizing that you need two different constraints — one about quantities and one about values — to solve for two unknowns.

Recognizing This Problem in the Wild

Look for these signal phrases that indicate a system of equations problem:

• "Two types" or "two kinds" of items with different values
• A total count AND a total value given
• "How many of each..." (asking for two unknown quantities)
• Different prices per unit for each category