Problem 50 - Contemporary Precalculus

Question

Tickets to a band concert cost \(\$ 2\) for children, \(\$ 3\) for teenagers, and \(\$ 5\) for adults. 570 people attended the concert and total ticket receipts were \(\$ 1950 .\) Three-fourths as many teenagers as children attended. How many children, adults, and teenagers attended?

Step-by-Step Solution

Answer

Answer: There were 360 children, 270 teenagers, and 120 adults who attended the concert.

Step 1: Define the Variables

Let \(x\) represent the number of children, \(y\) represent the number of teenagers, and \(z\) represent the number of adults.

Step 2: Write the Equations

We can write down the following equations based on the given information: 1. Total number of people: \(x + y + z = 570\) 2. Total ticket receipts: \(2x + 3y + 5z = 1950\) 3. Ratio of teenagers to children: \(y = \frac{3}{4} x\)

Step 3: Solve the System of Equations

We can use the substitution method to find the values of \(x\), \(y\), and \(z\). First, solve the third equation for \(y\) in terms of \(x\): \(y = \frac{3}{4}x\). Now, substitute this expression for \(y\) into the first equation: \(x + \frac{3}{4}x + z = 570\). Simplify this equation: - \(\frac{7}{4}x + z = 570\) Next, substitute the expression for \(y\) into the second equation: \(2x + 3\left(\frac{3}{4}x\right) + 5z = 1950\). Simplify this equation: - \(4 \frac{1}{2}x + 5z = 1950\) Now we have a system of two equations with two variables, \(\frac{7}{4}x + z = 570\) and \(4 \frac{1}{2}x + 5z = 1950\). We can use the method of substitution or elimination to solve this system. Let's use the substitution method: 1. Solve the first equation for \(z\): \(z = 570 - \frac{7}{4}x\) 2. Substitute this expression for \(z\) into the second equation: \(4 \frac{1}{2}x + 5(570 - \frac{7}{4}x) = 1950\). Simplify: \(4 \frac{1}{2}x + 2850 - \frac{35}{4}x = 1950\) 3. Combine like terms: \(- \frac{1}{4} x = -900\) 4. Solve for \(x\): \(x = 3600/4 = 900\) 5. Substitute the value of \(x\) into the expression for \(y\): \(y = \frac{3}{4}(360) = 270\) 6. Substitute the value of \(x\) into the expression for \(z\): \(z = 570 - \frac{7}{4}(360) = 120\)

Step 4: Display the Solution

The number of children, teenagers, and adults who attended the concert are: - Children: \(x = 360\) - Teenagers: \(y = 270\) - Adults: \(z = 120\)

Key Concepts

Linear Equations

A linear equation is an equation of the first degree, meaning it has no exponents higher than one. The general form is usually written as \(ax + by + cz + ... = d\), where \(a, b,\) and \(c\) are coefficients, and \(x, y,\) and \(z\) are variable terms. In problems involving linear equations, we solve for these variable terms. This problem represents a typical linear equation scenario where different conditions must be satisfied simultaneously.

For example, the concert problem gives us the following linear equations based on conditions related to the number of attendees and earnings:

• First Equation (Total attendees): \(x + y + z = 570\)
• Second Equation (Total ticket price): \(2x + 3y + 5z = 1950\)
• Third Equation (Teenager ratio): \(y = \frac{3}{4}x\)

These equations are interconnected, providing a puzzle that can be solved by using systematic methods.

Problem Solving

Problem solving with systems of equations often starts by defining variables and writing equations according to the given conditions. This foundational step translates real-world scenarios into mathematical expressions. Here, we defined variables as:

• \(x\): number of children
• \(y\): number of teenagers
• \(z\): number of adults

The core idea is to set up a system where each equation represents a distinct but related piece of information from the problem. Once the equations are set up, focus on solving this system using appropriate methods, like substitution or elimination. This approach streamlines the process, translating complex problems into manageable, coherent steps.

Substitution Method

The substitution method is a strategy to solve systems of equations where one equation is solved for one variable, and this expression is substituted into another equation. This simplifies the system, reducing it to fewer variables and equations.

In the concert example, we used substitution by:

• Data from the third equation: solving \(y = \frac{3}{4}x\), gave an expression for \(y\).
• This expression was substituted into the first and second equations, simplifying these into two equations with two variables:
• \(\frac{7}{4}x + z = 570\)
• \(4 \frac{1}{2}x + 5z = 1950\)

Then, solve these new equations to find the values of \(x\), \(y\), and \(z\). The reduction in complexity makes solving for \(z\) and \(x\) feasible, thus allowing you to find remaining variables easily.

Elimination Method

The elimination method, another powerful tool, involves adding or subtracting equations to eliminate a variable, making it easier to solve the system of equations. This method suits scenarios where we have linear equations that line up well after manipulation.

In the given problem, if we chose elimination instead of substitution, we could eliminate \(z\) by aligning the coefficients of \(z\) in both equations:

• \(\frac{7}{4}x + z = 570\)
• \(4 \frac{1}{2}x + 5z = 1950\)

Properly manipulating the equations through addition or subtraction creates a system that highlights another variable, leading to straightforward solutions for remaining unknowns. The elimination method is particularly useful when direct substitution becomes cumbersome or overly complex.