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A movie theater charges different prices for children and adults. On Friday, 10 adults and 25 children went to the movie and it cost $670. On Sunday 5 adults and 10 children went and it cost $290.

a) What equation would represent the cost of going to a movie on Friday? 10a + 25c = 670, 5a + 10c = 290

b) What equation would represent the cost of going to a movie on Sunday? 10a + 25c = 670, 5a + 10c = 290

c) What would be the best algebraic method to solve given how the equations are set up? Substitution because one of the equations already has a variable solved for, Elimination because the equations are already lines up nicely - you could just multiply the Sunday equation by -2 to eliminate the 'a' variable

d) What is the cost of an adult ticket? 18, 58, 29, 22

1 answer

Let's break down the problem step-by-step.

a) Equation representing the cost of going to a movie on Friday:

The cost for Friday can be modeled with the equation:

  • 10a + 25c = 670

Here, \(a\) represents the cost of an adult ticket, and \(c\) represents the cost of a child ticket.

b) Equation representing the cost of going to a movie on Sunday:

The cost for Sunday can be modeled with the equation:

  • 5a + 10c = 290

c) Best algebraic method to solve the system:

To solve the system of equations:

  • From the two equations, we have:
  1. 10a + 25c = 670 (Equation 1)
  2. 5a + 10c = 290 (Equation 2)

A good method to solve would be Elimination because the second equation can be easily manipulated. You can multiply the second equation by -2 to align the coefficients of \(a\):

Multiply Equation 2 by -2:

  • -10a - 20c = -580 (Equation 3)

Now we can add Equation 1 and Equation 3:

  • \( (10a + 25c) + (-10a - 20c) = 670 - 580 \)
  • \( 5c = 90 \)
  • \( c = 18 \)

Now, substitute \(c = 18\) back into one of the original equations (let's use Equation 2): \[ 5a + 10(18) = 290 \] \[ 5a + 180 = 290 \] \[ 5a = 110 \] \[ a = 22 \]

d) Cost of an adult ticket:

From our calculations, we find the cost of an adult ticket is:

  • 22

Thus, to summarize:

  • a) 10a + 25c = 670
  • b) 5a + 10c = 290
  • c) Elimination method is preferable.
  • d) The cost of an adult ticket is 22.