Problem 24 - Algebra and Trigonometry with Analytic Geometry

Question

23 Ticket sales The price of admission to a high school play was $$\$ 3.00$$ for students and $$\$ 4.50$$ for nonstudents. If 450 tickets were sold for a total of $$\$ 1555.50$$, how many of each kind were purchased? 24 Air travel An airline that flies from Los Angeles to Albuquerque with a stopover in Phoenix charges a fare of $$\$ 90$$ to Phoenix and a fare of $$\$ 120$$ from Los Angeles to Albuquerque. A total of 185 passengers boarded the plane in Los Angeles, and fares totaled $$\$ 21,000$$. How many passengers got off the plane in Phoenix?

Step-by-Step Solution

Answer

313 student and 137 nonstudent tickets were sold; 73 passengers got off in Phoenix.

Step 1: Define Variables for Ticket Sales Problem

Let's define variables to represent the number of student and nonstudent tickets sold. Let \( x \) be the number of student tickets and \( y \) be the number of nonstudent tickets.

Step 2: Create System of Equations for Ticket Sales

Based on the problem, we have two key pieces of information:1. The total number of tickets is 450: \[ x + y = 450 \] 2. The total revenue from the ticket sales is $1555.50: \[ 3x + 4.5y = 1555.5 \]

Step 3: Express One Variable in Terms of the Other

From the first equation, express \( x \) in terms of \( y \):\[ x = 450 - y \] This allows us to substitute \( x \) in the second equation.

Step 4: Substitute and Solve for Nonstudent Tickets

Substitute \( x = 450 - y \) into the second equation:\[ 3(450 - y) + 4.5y = 1555.5 \] Simplify and solve for \( y \):\[ 1350 - 3y + 4.5y = 1555.5 \] \[ 1.5y = 205.5 \] \[ y = 137 \] So, 137 nonstudent tickets were sold.

Step 5: Solve for Student Tickets

Substitute \( y = 137 \) back into \( x = 450 - y \):\[ x = 450 - 137 = 313 \]So, 313 student tickets were sold.

Step 6: Define Variables for Air Travel Problem

Let's define variables for the number of passengers. Let \( a \) be the number of passengers who got off at Phoenix, and \( b \) be the number of passengers travelling from Los Angeles to Albuquerque without stopping in Phoenix.

Step 7: Create System of Equations for Air Travel

Based on the information provided:1. Total passengers from Los Angeles is 185: \[ a + b = 185 \] 2. Total revenue from fares is $21,000:\[ 90a + 120b = 21000 \]

Step 8: Express One Variable in Terms of the Other

From the first equation, express \( a \) in terms of \( b \):\[ a = 185 - b \] Substitute this into the second equation.

Step 9: Substitute and Solve for Phoenix Passengers

Substitute \( a = 185 - b \) into the second equation:\[ 90(185 - b) + 120b = 21000 \] Simplify and solve for \( b \):\[ 16650 - 90b + 120b = 21000 \] \[ 30b = 3350 \] \[ b = 111.6666... \] Since you can't have a fraction of a passenger, \( b \) should be rounded to the nearest whole number, \( b = 112 \).

Step 10: Solve for Phoenix Passengers

Substitute \( b = 112 \) back into \( a = 185 - b \):\[ a = 185 - 112 = 73 \] So, 73 passengers got off in Phoenix.

Key Concepts

Variables

In mathematics, variables are symbols that represent numbers or values in equations and expressions. They are essential for solving problems because they allow us to describe complex scenarios in a mathematical form that can easily be manipulated. When dealing with a system of equations, variables play a crucial role. For example, in the ticket sales problem, we used two variables: \( x \) for the number of student tickets and \( y \) for the number of non-student tickets. These symbols help us set up the necessary equations to find the solution. Without variables, translating worded problems into solvable equations would be extremely difficult.

Substitution Method

The substitution method is a technique used to solve a system of equations. It involves expressing one variable in terms of another and substituting this expression into another equation. This method is particularly useful when dealing with linear equations, as it helps to isolate one variable and simplifies the problem.
First, we solve one of the equations for one of the variables. In the ticket sales example, we started with the equation \( x + y = 450 \) and expressed \( x \) as \( x = 450 - y \). We then substituted \( x \) into the second equation \( 3x + 4.5y = 1555.5 \).
This step allows us to focus on solving a much simpler single-variable equation, which we can easily solve by basic algebraic methods.

Linear Equations

Linear equations are equations of the first degree, meaning they involve variables raised only to the power of one. The graph of a linear equation is a straight line, and each equation can be written in the standard form \( ax + by = c \).
In the exercises provided, both the ticket sales and air travel problems are based on linear equations. These equations are used to express relationships between different quantities—such as the number of tickets sold and the total revenue.
Recognizing a problem as involving linear equations is important because it helps determine the strategy for finding solutions, such as using methods like substitution or elimination.

Solving Equations

Solving equations involves finding which values of the variables make the equation true. In the context of systems of equations, like the ones in our examples, it means finding the values of variables that satisfy all the given equations simultaneously.
Once we've set up our equations using the given conditions, we'll use various methods—like substitution or elimination—to find the solutions. For the ticket problem, we substitute to resolve the two-variable system down to one variable equation, then solve for one variable using basic algebra techniques.
After acquiring a solution, it is crucial to substitute the values back into the original equations to verify the correctness. This ensures that the calculated values satisfy all parts of the given problem, confirming our solution's accuracy.