Problem 63 - College Algebra with Corequisite Support

Question

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer's Rule. A movie theatre sold tickets to three movies. The tickets to the first movie were \(\$ 5,\) the tickets to the second movie were \(\$ 11\), and the third movie was \(\$ 12\). 100 tickets were sold to the first movie. The total number of tickets sold was \(642,\) for a total revenue of \(\$ 6,774\). How many tickets for each movie were sold?

Step-by-Step Solution

Answer

100 tickets for the first movie, 232 for the second, and 312 for the third.

Step 1: Define the Variables

Let \( x \) represent the number of tickets sold for the first movie, \( y \) the number for the second movie, and \( z \) for the third movie.

Step 2: Set Up the Equations

We know 100 tickets were sold for the first movie, so \( x = 100 \). Additionally, the total number of tickets sold is 642, yielding the equation \( x + y + z = 642 \). The total revenue is \$ 6,774, described by \( 5x + 11y + 12z = 6774 \).

Step 3: Substitute Known Values

Since we know \( x = 100 \), substitute it into the equations: \( 100 + y + z = 642 \) which simplifies to \( y + z = 542 \) and \( 5(100) + 11y + 12z = 6774 \) which simplifies to \( 11y + 12z = 6274 \).

Step 4: Write the System of Equations

Our system of equations is now: \[ \begin{align*} y + z &= 542 \ 11y + 12z &= 6274 \end{align*} \]

Step 5: Apply Cramer's Rule

Cramer's Rule requires calculation of determinants.The coefficients from the system are:\[ A = \begin{pmatrix} 1 & 1 \ 11 & 12 \end{pmatrix} \]Calculate the determinant of \( A \):\[ \text{det}(A) = (1)(12) - (1)(11) = 1 \]

Step 6: Determine D_y and D_z

Replace the first column of \( A \) for \( D_y \):\[ D_y = \begin{pmatrix} 542 & 1 \ 6274 & 12 \end{pmatrix} \], so\[ \text{det}(D_y) = (542)(12) - (1)(6274) = 6506 - 6274 = 232 \]Replace the second column for \( D_z \):\[ D_z = \begin{pmatrix} 1 & 542 \ 11 & 6274 \end{pmatrix} \], so\[ \text{det}(D_z) = (1)(6274) - (11)(542) = 6274 - 5962 = 312 \]

Step 7: Find Solutions for y and z

Using Cramer's Rule:\[ y = \frac{\text{det}(D_y)}{\text{det}(A)} = \frac{232}{1} = 232 \]\[ z = \frac{\text{det}(D_z)}{\text{det}(A)} = \frac{312}{1} = 312 \]

Step 8: Conclusion

The solution is: \( x = 100 \), \( y = 232 \), and \( z = 312 \). Thus, 100 tickets were sold for the first movie, 232 for the second, and 312 for the third.

Key Concepts

System of Linear Equations

A system of linear equations consists of multiple linear equations that share a set of variables. Solving these systems allows us to find values for the variables that make all the equations true simultaneously. In the context of our movie theatre problem, our variables represent the number of tickets sold for each movie.
For instance, in our example, we defined:

• \( x \) as the number of tickets sold for the first movie, which we know is 100.
• \( y \) and \( z \) as the number of tickets for the second and third movies respectively.

To solve, we set up the following linear equations based on given conditions:

• The total number of tickets equation: \( x + y + z = 642 \).
• The total revenue equation: \( 5x + 11y + 12z = 6774 \).

By substituting the known value of \( x = 100 \) into these equations, we simplify and reduce the system to two equations with two variables. This enables easier solving using techniques like substitution or Cramer's Rule.

Determinants

Determinants are mathematical tools used to analyze square matrices, helping in solving systems of linear equations. They play a critical role in Cramer's Rule, which provides a method for finding a unique solution to the system of linear equations using determinants.
In our problem, we first form a coefficient matrix \( A \). This matrix includes the coefficients from the reduced system of equations:

• From \( y + z = 542 \) and \( 11y + 12z = 6274 \), the matrix \( A \) is \( \begin{pmatrix} 1 & 1 \ 11 & 12 \end{pmatrix} \).

To apply Cramer's Rule, we calculate the determinant of \( A \):

• \( \text{det}(A) = (1)(12) - (1)(11) = 1 \).

The determinant here tells us if the system has a unique solution, infinite solutions, or no solution. A non-zero determinant, as in our case, indicates a unique solution.

Algebra Problem Solving

Algebra problem solving involves manipulating equations to find unknown values. It often requires a step-by-step approach, combining various algebraic techniques.

• First, clearly define variables. Identifying what each variable represents helps in setting up the correct equations.
• Rearrange or substitute equations. Solving our movie ticket example involved substituting \( x = 100 \) into the system, reducing the problem to manageable equations with just two unknowns, \( y \) and \( z \).
• Identify and choose a suitable method. In this context, we employed Cramer's Rule, which leverages determinants, providing an efficient strategy for solving the equations.
• Calculate to achieve the final solution. Perform determinant calculations and solve for the remaining variables \( y \) and \( z \). The tidy use of algebraic concepts gives the final answer \( y = 232 \) and \( z = 312 \).

Everything culminates in verifying the results by adding the solutions back into the equations to ensure they satisfy the original conditions, thus confirming the solution's accuracy.