A theater sold adult and student tickets for two performances. For the afternoon show, 3a + 7s = 59. For...

1. TRANSLATE the problem information

• Given information:
  • Afternoon show: 3 adult tickets + 7 student tickets = $59
  • Evening show: 5 adult tickets + 4 student tickets = $60
  • \(\mathrm{a}\) = price of one adult ticket, \(\mathrm{s}\) = price of one student ticket

• This translates to the system:
  • \(\mathrm{3a + 7s = 59}\)
  • \(\mathrm{5a + 4s = 60}\)

2. INFER the solution strategy

• We need to solve for both variables to find their product as
• Since we have two equations with two unknowns, we can use elimination method
• Looking at the coefficients, we can eliminate 'a' by making them equal

3. SIMPLIFY by setting up elimination

• Multiply first equation by 5: \(\mathrm{15a + 35s = 295}\)
• Multiply second equation by 3: \(\mathrm{15a + 12s = 180}\)
• Now both equations have 15a, so we can eliminate this variable

4. SIMPLIFY to solve for s

• Subtract the second equation from the first:
  \(\mathrm{(15a + 35s) - (15a + 12s) = 295 - 180}\)
• \(\mathrm{23s = 115}\)
• \(\mathrm{s = 5}\)

5. SIMPLIFY to solve for a

• Substitute \(\mathrm{s = 5}\) into either original equation. Using \(\mathrm{5a + 4s = 60}\):
• \(\mathrm{5a + 4(5) = 60}\)
• \(\mathrm{5a + 20 = 60}\)
• \(\mathrm{5a = 40}\)
• \(\mathrm{a = 8}\)

6. Find the final answer

• \(\mathrm{as = 8 \times 5 = 40}\)

Answer: 40

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to set up the correct system of equations from the word problem.

They might write incorrect equations like "\(\mathrm{3a + 7s + 5a + 4s = 59 + 60}\)" (combining all information into one equation) or mix up which numbers correspond to which show. This fundamental translation error prevents them from even beginning the solution process correctly, leading to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors during the elimination process.

Common mistakes include sign errors when subtracting equations, incorrect multiplication when setting up elimination, or arithmetic errors when solving for variables. For example, they might get \(\mathrm{23s = 115}\) but calculate \(\mathrm{s = 4}\) instead of \(\mathrm{s = 5}\), leading them to \(\mathrm{a = 10}\) and \(\mathrm{as = 40}\)... wait, that still gives 40. Let me think of a better example. If they make an error and get \(\mathrm{s = 4}\), then substituting: \(\mathrm{5a + 4(4) = 60}\), so \(\mathrm{5a = 44}\), \(\mathrm{a = 8.8}\), giving \(\mathrm{as = 35.2}\), which would lead them to round or guess among available choices.

The Bottom Line:

This problem requires students to bridge the gap between word problems and algebraic systems, then execute multiple algebraic steps flawlessly. The challenge lies not just in translating the scenario correctly, but in maintaining accuracy through the multi-step elimination process.