A train ticket costs $[math: x] for an adult and $[math: y] for a child. The total cost of 6 adult tickets and 10 child tickets is $[math: 56].The cost of an adult ticket is 3 times the cost of a child ticket.The correct set of simultaneous equations representing the above information are

Question

A train ticket costs $[math: x] for an adult and $[math: y] for a child. The total cost of 6  adult tickets and 10 child tickets is $[math: 56].The cost of an adult ticket is 3 times the cost of a child ticket.The correct set of simultaneous equations representing the above information are

BlackTom AI · Accepted Answer

Let x be the price of an adult ticket and y be the price of a child ticket.

1. Relevant knowledge points and concepts
- Forming linear equations from verbal descriptions: multiply price per item by quantity to get total cost.
- Translating comparative statements into algebraic relations (e.g., "A is 3 times B" means A = 3B).
- Solving a simple pair of linear equations by substitution to check consistency.

2. Step-by-step reasoning
- The total cost of 6 adult tickets and 10 child tickets is $56. That becomes 6·x + 10·y = 56.
- “The cost of an adult ticket is 3 times the cost of a child ticket” translates to x = 3y (adult equals three times child).

3. Calculations (all intermediate steps)
Substitute x = 3y into the total-cost equation:
6x + 10y = 56
6(3y) + 10y = 56
18y + 10y = 56
28y = 56
y = 56 / 28
y = 2
Then x = 3y = 3·2 = 6
So adult ticket $6 and child ticket $2, which satisfies 6·6 + 10·2 = 36 + 20 = 56.

4. Why the correct answer is correct
The pair 6x + 10y = 56 and x = 3y exactly encodes the two given facts: the correct coefficient for 6 adults is 6x, for 10 children is 10y, and the comparative relation is adult = 3 × child. Solving yields sensible prices that check the total.

5. Why the other options are incorrect (brief)
- A (6x + 10y = 56 and y = 3x): reverses the comparison, claiming child = 3 × adult, which contradicts the statement.
- B (10x + 6y = 56 and x = 3y): swaps the quantities (treats 10 adults and 6 children), so the cost equation is wrong even though x = 3y is correct.
- C (10x + 6y = 56 and y = 3x): both the quantity assignment and the comparison are wrong.

6. Conclusion
The correct set of simultaneous equations is 6x + 10y = 56 and x = 3y, which matches answer D.

类似问题

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