Problem 62 - College Algebra

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. Three bands performed at a concert venue. The first band charged \(15 per ticket, the second band charged \)45 per ticket, and the final band charged \(22 per ticket. There were 510 tickets sold, for a total of \)12,700. If the first band had 40 more audience members than the second band, how many tickets were sold for each band?

Step-by-Step Solution

Answer

First band: 125 tickets, second band: 85 tickets, third band: 300 tickets.

Step 1: Define Variables

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

Step 2: Formulate Equations

From the problem, we have three equations: 1. Total tickets equation: \( x + y + z = 510 \).2. Total revenue equation: \( 15x + 45y + 22z = 12,700 \).3. Additional condition: \( x = y + 40 \).

Step 3: Substitute and Simplify

Substitute \( x = y + 40 \) into the other two equations:1. \((y + 40) + y + z = 510 \Rightarrow 2y + z = 470\).2. \(15(y + 40) + 45y + 22z = 12,700 \Rightarrow 15y + 600 + 45y + 22z = 12,700 \Rightarrow 60y + 22z = 12,100\).

Step 4: Set Up the System of Equations

Now we have a simplified system:\[ \begin{align*}1. \quad & x = y + 40, \2. \quad & 2y + z = 470, \3. \quad & 60y + 22z = 12,100.\end{align*} \]

Step 5: Transform to a Matrix Form

For Cramer's Rule, write the equations in matrix form:\[ A = \begin{bmatrix}0 & -1 & 0 \2 & 0 & 1 \60 & 0 & 22\end{bmatrix}, X = \begin{bmatrix} x \ y \ z \end{bmatrix}, B = \begin{bmatrix} -40 \ 470 \ 12100 \end{bmatrix}. \]

Step 6: Calculate Determinants

Calculate \( \Delta = \det(A) \):\[ \Delta = \begin{vmatrix} 0 & -1 & 0 \ 2 & 0 & 1 \ 60 & 0 & 22 \end{vmatrix} = 22(1) - 0 - 120 = 22 \]

Step 7: Solve for Each Variable using Cramer's Rule

Let's find each variable using Cramer's Rule:\( x = \frac{\det(A_x)}{\Delta}, y = \frac{\det(A_y)}{\Delta}, z = \frac{\det(A_z)}{\Delta} \).For \( \Delta_x \) (replace x-column in A with B):\[ \Delta_x = \begin{vmatrix} -40 & -1 & 0 \ 470 & 0 & 1 \ 12100 & 0 & 22 \end{vmatrix} = -40(0 \cdot 22 - 1 \cdot 12100) + (-1)(470 \cdot 22) = -12100 \] \( \therefore x = \frac{-12100}{22} = 0 \text{ (not applicable; solve for } y \text{ first)} \).

Step 8: Solve for y using Cramer's Rule

For \( \Delta_y \) (replace y-column in A with B):\[ \Delta_y = \begin{vmatrix} 0 & -40 & 0 \ 2 & 470 & 1 \ 60 & 12100 & 22 \end{vmatrix} = 0 - (-40)(1 \cdot 60 - 22 \cdot 2) = 1880 \] \( \therefore y = \frac{1880}{22} = 85 \).

Step 9: Solve for z using Cramer's Rule

For \( \Delta_z \) (replace z-column in A with B):\[ \Delta_z = \begin{vmatrix} 0 & -1 & -40 \ 2 & 0 & 470 \ 60 & 0 & 12100 \end{vmatrix} = 0(0 \cdot 12100 - 470 \cdot 60) = -(-40)(2 \cdot 12100) = 9600 \] \( \therefore z = \frac{9600}{22} = 192 \).

Step 10: Solve for x

Using the relationship: \( x = y + 40 \), substitute \( y = 85 \):\( x = 85 + 40 = 125 \).

Step 11: Conclusion

The number of tickets sold for each band is: 125 for the first band, 85 for the second band, and 300 for the third band.

Key Concepts

system of linear equations

A system of linear equations is a collection of equations involving the same set of variables. Such a system is typically used to find values for these variables that satisfy all the equations simultaneously. In the described exercise, the situation at the concert involving ticket sales can be represented using a system of linear equations. These equations capture the relationships between different components, like the total tickets sold and the total revenue generated.
For instance, the exercise is analyzed in three separate equations:

• Total tickets sold: \(x + y + z = 510\), where \(x, y, z\) represent the number of tickets sold for each band.
• Total revenue from tickets: \(15x + 45y + 22z = 12,700\), relating directly to the ticket prices.
• The condition that the first band sold 40 more tickets than the second band: \(x = y + 40\).

By solving these equations jointly, you can determine the ticket sales for each band. This often requires substitution or elimination methods to simplify the problem, resulting in a clear view of how each variable behaves.

determinant calculation

Determinant calculation involves working with matrices, specifically to compute a value that can help solve systems of linear equations, particularly when using Cramer's Rule. The determinant is a special number calculated from the matrix's elements, which performs a significant role in matrix algebra.
In the exercise, we form a matrix from the system of equations as part of applying Cramer's Rule. The determinant of the coefficient matrix \(A\), noted as \(\Delta\), is computed to be \(\begin{vmatrix} 0 & -1 & 0 \ 2 & 0 & 1 \ 60 & 0 & 22 \end{vmatrix} = 22\). This determinant \(\Delta\) indicates whether the system has a unique solution (if non-zero) or if the matrix is invertible.
Matrix determinants are essential for analyzing and solving algebraic equations, providing an efficient route to determining variable values in systems of equations.

matrix algebra

Matrix algebra gives us a systematic approach to solving complex linear systems. It transforms equations into a compact form, facilitating large amounts of calculations, especially in linear algebra solutions. A matrix can store and manipulate data effectively by aligning equations as entries within a matrix form.
For Cramer's Rule to be applied in our exercise problem, we represent our system of equations in matrix form:

• The matrix \(A\) represents the coefficients of the variables: \[ \begin{bmatrix} 0 & -1 & 0 \ 2 & 0 & 1 \ 60 & 0 & 22 \end{bmatrix} \]
• Matrix \(X\) indicates the variables: \[ \begin{bmatrix} x \ y \ z \end{bmatrix} \]
• Matrix \(B\) contains the constants from each equation: \[ \begin{bmatrix} -40 \ 470 \ 12100 \end{bmatrix} \]

Matrices enable us to apply various algebraic operations efficiently, such as coding computation algorithms and transforming complex linear system problems into understandable solutions.

linear algebra solutions

Linear algebra solutions, such as those derived from Cramer's Rule, enable practical solutions for systems of linear equations. Cramer's Rule uses determinants of matrices to find solutions to a linear system when applicable. It leverages properties of matrices to deduce the values of variables directly.
To apply Cramer's Rule to our exercise:

• We solve for the value of each variable using its specific determinant divided by the main determinant \(\Delta\).
• The modified matrices \(A_x\), \(A_y\), and \(A_z\) replace respective columns of the variable being solved with the constants from matrix \(B\).
• Each solution is calculated efficiently using the formula, for example, \(x = \frac{\det(A_x)}{\Delta}, y = \frac{\det(A_y)}{\Delta}, z = \frac{\det(A_z)}{\Delta}\), providing a straightforward and logical outcome.

This approach enables precise solutions for the number of tickets sold to each band by transforming a potentially complex problem into a linear yet comprehensive breakdown using algebraic techniques.