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Difficulty: MediumSystems of Linear Equations

A community theater sold adult tickets and student tickets for its opening weekend performances. On Friday, the theater sold 2020 adult tickets and 4040 student tickets for a total of $760\$760. On Saturday, the theater sold 3535 adult tickets and 3030 student tickets for a total of $930\$930. What is the price, in dollars, of one adult ticket?

  1. A
    10
  2. B
    16
  3. 18Answer
  4. D
    30

Answer

The price of one adult ticket is 18 dollars.
The price of one adult ticket is 1818. By setting up the system of linear equations representing the total sales on Friday (20x+40y=76020x + 40y = 760) and Saturday (35x+30y=93035x + 30y = 930), where xx is the adult ticket price and yy is the student ticket price, we can simplify them to x+2y=38x + 2y = 38 and 7x+6y=1867x + 6y = 186. Multiplying the first simplified equation by 33 gives 3x+6y=1143x + 6y = 114. Subtracting this equation from 7x+6y=1867x + 6y = 186 eliminates yy and yields 4x=724x = 72, which simplifies to x=18x = 18.

Step-by-Step Solution

1
Define variables and write the system of linear equations based on the given information.
Let xx represent the price of an adult ticket and yy represent the price of a student ticket. The system of equations is:
20x+40y=76020x + 40y = 760
35x+30y=93035x + 30y = 930
Variables represent the unknown quantities, and the equations relate these quantities to the total sales on Friday and Saturday.
2
Simplify both equations by dividing by their greatest common divisors.
Divide the first equation by 2020:
x+2y=38x + 2y = 38
Divide the second equation by 55:
7x+6y=1867x + 6y = 186
Simplifying equations reduces the coefficient values and minimizes the chance of arithmetic errors in later steps.
3
Use the elimination method to solve for xx. Multiply the first simplified equation by 33 to align the yy-coefficients.
3(x+2y)=3(38)⇒3x+6y=1143(x + 2y) = 3(38) \Rightarrow 3x + 6y = 114
Matching the coefficients of yy allows us to eliminate the variable yy by subtracting the equations.
4
Subtract the equation obtained in Step 3 from the second simplified equation.
(7x+6y)−(3x+6y)=186−114⇒4x=72(7x + 6y) - (3x + 6y) = 186 - 114 \Rightarrow 4x = 72
Subtracting the equations eliminates yy, leaving a single linear equation in terms of xx.
5
Solve for xx by dividing both sides of the equation by 44.
x=18x = 18
Isolating xx provides the price of one adult ticket.

Key Concept

Solving a system of linear equations using substitution or elimination.

Alternative Method

Instead of using elimination on the simplified equations, you can use substitution. Express xx in terms of yy using the first simplified equation: x=38−2yx = 38 - 2y. Substitute this expression into the second equation: 7(38−2y)+6y=1867(38 - 2y) + 6y = 186. Expanding this gives 266−14y+6y=186266 - 14y + 6y = 186, which simplifies to 266−8y=186266 - 8y = 186. Subtracting 266266 from both sides gives −8y=−80-8y = -80, which solves to y=10y = 10. Finally, substitute y=10y = 10 back into x=38−2yx = 38 - 2y to find x=38−2(10)=18x = 38 - 2(10) = 18.
Estimated Time:1m 30s
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Topics

Systems of Linear Equations

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