Problem 31 - Precalculus

Question

The price for admission into the Stitz-Zeager Sasquatch Museum and Research Station is $$\$ 15$$ for adults and $$\$ 8$$ for kids 13 years old and younger. When the Zahlenreich family visits the museum their bill is $$\$ 38$$ and when the Nullsatz family visits their bill is $$\$ 39 .$$ One day both families went together and took an adult babysitter along to watch the kids and the total admission charge was $$\$ 92$$. Later that summer, the adults from both families went without the kids and the bill was $$\$ 45 .$$ Is that enough information to determine how many adults and children are in each family? If not, state whether the resulting system is inconsistent or consistent dependent. In the latter case, give at least two plausible solutions.

Step-by-Step Solution

Answer

Yes, it's consistent dependent. One solution is 2 adults and 1 child for Zahlenreich, 1 adult and 3 children for Nullsatz.

Step 1: Identifying Variables

Let's define variables for the number of adults and children in each family. Let \( a_1 \) and \( c_1 \) represent the number of adults and children in the Zahlenreich family, respectively. Similarly, let \( a_2 \) and \( c_2 \) represent the number of adults and children in the Nullsatz family.

Step 2: Setting Up Equations

We can set up equations based on the information given:1. For the Zahlenreich family: \( 15a_1 + 8c_1 = 38 \).2. For the Nullsatz family: \( 15a_2 + 8c_2 = 39 \).3. Both families with an adult babysitter and all kids: \( 15(a_1 + a_2 + 1) + 8(c_1 + c_2) = 92 \).4. Adults from both families: \( 15(a_1 + a_2) = 45 \).

Step 3: Solving The Adults Equation

From the adults' equation, \( 15(a_1 + a_2) = 45 \), divide both sides by 15 to find the total number of adults from both families:\[ a_1 + a_2 = 3 \]

Step 4: Revisiting Total Equation

Substitute \( a_1 + a_2 = 3 \) into the babysitter equation:\[ 15(3 + 1) + 8(c_1 + c_2) = 92 \]This simplifies to:\[ 60 + 8(c_1 + c_2) = 92 \]Subtract 60 from both sides:\[ 8(c_1 + c_2) = 32 \]Divide by 8:\[ c_1 + c_2 = 4 \]

Step 5: Determining Family Equations

We now have:1. \( a_1 + a_2 = 3 \)2. \( c_1 + c_2 = 4 \)These equations are consistent with the previously obtained equations. Let's look at the individual family equations:- Zahlenreich: \( 15a_1 + 8c_1 = 38 \)- Nullsatz: \( 15a_2 + 8c_2 = 39 \)

Step 6: Finding Consistent Solutions

To find solutions, assume various values to satisfy both family equations. Start with potential configurations of \( a_1 \) and \( c_1 \) satisfying \( a_1 + a_2 = 3 \) and \( c_1 + c_2 = 4 \).1. Possible solution: \( a_1 = 1, c_1 = 2 \) which gives \( 15 \times 1 + 8 \times 2 = 31 \) (inconsistent).2. Another solution: \( a_1 = 2, c_1 = 1 \) gives \( 15 \times 2 + 8 \times 1 = 38 \) (consistent). Then \( a_2 = 1, c_2 = 3 \) gives \( 15 \times 1 + 8 \times 3 = 39 \) (consistent).Hence, one plausible division is \( a_1 = 2, c_1 = 1, a_2 = 1, c_2 = 3 \). The resulting solutions are consistent with the total equations.

Key Concepts

Consistent and Dependent Systems

When working with systems of equations, understanding whether the system is consistent or inconsistent is crucial. A consistent system has at least one solution. If it has infinitely many solutions, it may also be classified as dependent.
Let's examine the scenario from the museum visit problem.
In this exercise, the equations provided describe the situation mathematically:

• For the Zahlenreich family: \( 15a_1 + 8c_1 = 38 \)
• For the Nullsatz family: \( 15a_2 + 8c_2 = 39 \)
• Together with an adult babysitter: \( 15(a_1 + a_2 + 1) + 8(c_1 + c_2) = 92 \)
• Only adults: \( 15(a_1 + a_2) = 45 \)

By solving these, we find that the system is consistent, with solutions that satisfy all equations. This means the family compositions were determined correctly, showing an understanding of consistent systems in action.

Substitution Method

The substitution method is a classic technique used to solve systems of equations, typically when one equation can be easily solved for one variable. Once substituted into the other equation(s), it simplifies finding common solutions.
In the museum problem, we see its application by substituting derived totals into the equations. After determining there are three adults in total \( a_1 + a_2 = 3 \), and combining this into the babysitter equation, the task simplifies. The value for \( c_1 + c_2 = 4 \) can be found using this approach.
This ultimately allows us to test various combinations of \( a_1, a_2, c_1, \) and \( c_2 \) that still satisfy all conditions. The substitution method showcases its efficacy in gradually reducing complexity in solving systems.

Solving Linear Equations

Solving linear equations forms the backbone of many real-world mathematical applications. Linear equations are expressions that depict a straight line when graphed, typically of the form \( ax + by = c \). The challenge often lies in managing multiple variables and ensuring all criteria are met.
In this example, each family has a unique equation based on admission pricing, and solving these means deriving values that work across multiple contexts:

• Zahlenreich family: addresses \( a_1 \) and \( c_1 \)
• Nullsatz family: focuses on \( a_2 \) and \( c_2 \)

By systematically addressing each equation, you ensure every variable aligns with the real-life scenario presented. This step-by-step examination helps understand both the process and precision required in manipulating and verifying linear equations.

Step-by-Step Problem Solving

Approaching problems with a step-by-step strategy ensures clarity and successful problem resolution. This methodical approach is beneficial, especially for complex systems, ensuring no detail is overlooked.
In solving the problem about family visits to the museum, this approach entailed identifying variables, setting up equations, and solving them one at a time. It involved multiple checks to verify the solutions meet all conditions:

• Initially defining \( a_1, a_2, c_1, \) and \( c_2 \) clearly.
• Organizing equations to represent each scenario and condition given.
• Applying simpler derived solutions to reduce overall complexity.
• Checking multiple scenarios for possible solutions that fit all results.

This method gives students a reliable process to tackle systems of equations, both in this context and broader mathematical settings.