Problem 53 - A Graphical Approach to Precalculus with Limits

Question

Three students buy different combinations of tickets for a baseball game. The first student buys 2 senior, 1 adult, and 2 student tickets for \(\$ 51 .\) The second student buys 1 adult and 5 student tickets for \(\$ 55 .\) The third student buys 2 senior, 2 adult, and 7 student tickets for \(\$ 75\). If possible, find the price of each type of ticket. Interpret your answer.

Step-by-Step Solution

Answer

There is no solution due to a contradiction in the equations.

Step 1: Set Up Equations

We have to set up equations based on the different ticket combinations and their total prices. Let \( s \) be the price of a senior ticket, \( a \) be the price of an adult ticket, and \( t \) be the price of a student ticket. From the student purchases, we have the following equations: 1. \( 2s + 1a + 2t = 51\) \(\text{(Equation 1)}\)2. \( 0s + 1a + 5t = 55\) \(\text{(Equation 2)}\)3. \( 2s + 2a + 7t = 75\) \(\text{(Equation 3)}\)

Step 2: Solve for One Variable

We can solve Equation 2 for \( a \):\[ 1a + 5t = 55 \]So, \[ a = 55 - 5t \]

Step 3: Substitute Back

Substitute \( a = 55 - 5t \) into Equations 1 and 3:- For Equation 1:\[ 2s + (55 - 5t) + 2t = 51 \]This simplifies to:\[ 2s - 3t = -4 \] \( \text{(Equation 4)}\)- For Equation 3:\[ 2s + 2(55 - 5t) + 7t = 75 \]This simplifies to:\[ 2s - 3t = -35 \] \( \text{(Equation 5)}\)

Step 4: Solve the System of Linear Equations

Solve Equations 4 and 5:- Equation 4: \( 2s - 3t = -4 \)- Equation 5: \( 2s - 3t = -35 \)\( -4 eq -35 \), so there is a contradiction, indicating no solution.

Step 5: Interpret the Result

The system of equations has a contradiction, meaning there is no set of values for \( s \), \( a \), and \( t \) that satisfy all the given conditions. This indicates that there is probably an error or inconsistency in the problem statement or the data provided.

Key Concepts

Linear Algebra

Linear algebra is an essential area of mathematics that deals with vectors and linear equations. It is widely used to understand and solve systems of linear equations like the one in our exercise. When faced with multiple linear equations, linear algebra provides a systematic way to find solutions that satisfy all equations at once.

In the given exercise, each equation represents a different student's ticket purchase. By setting up a system of equations, we can leverage linear algebra techniques to understand how these different ticket types relate to each other in terms of price:

• Two senior, one adult, and two student tickets: this leads to Equation 1.
• One adult and five student tickets: this leads to Equation 2.
• Two senior, two adult, and seven student tickets: this leads to Equation 3.

By organizing these ticket combinations into equations, we delve into linear algebra, specifically systems of equations, to find potential solutions or to identify inconsistencies.

Equation Solving

Solving equations is a fundamental part of mathematics, particularly when dealing with systems of equations. In this exercise, after setting up the equations from the ticket purchases, we attempt to find values for each type of ticket that balance all equations.

We modify Equation 2 to express the price of adult tickets, which gives us a more straightforward variable to substitute into other equations:

• Express the variable 'a' as \(a = 55 - 5t\).

This substitution technique allows us to reduce the number of variables and solve the remaining equations more easily. However, upon substituting and simplifying, we discover a contradiction, meaning there isn’t a consistent solution.

This exercise highlights how careful manipulation and substitution can simplify complex systems but also shows that sometimes systems are not solvable due to inconsistencies or mistakes in the given data.

Mathematical Modeling

Mathematical modeling involves using equations to represent real-world scenarios. In this exercise, the scenario consists of ticket purchases by different students for a baseball game. Each purchase is translated into a linear equation, modelling the situation mathematically.

The art of mathematical modeling is in accurately translating real-world details into mathematical language. Here, we had:

• The ticket purchases act as inputs.
• The equations translate these inputs mathematically to find prices.

But, sometimes, as with our exercise, there can be hidden assumptions or errors in the initial problem setup. The inconsistency found in the solution (no valid set of ticket prices fulfills all equations) often indicates that either there’s a mistake in data or assumptions. It's crucial in mathematical modeling to not only compute answers but also to critically analyze results and ensure they align logically with the real-world situation being modeled.