Problem 31 - College Algebra

Question

At a college production of Evita, 400 tickets were sold. The ticket prices were \(\$ 8, \$ 10,\) and \(\$ 12,\) and the total income from ticket sales was \(\$ 3700 .\) How many tickets of each type were sold if the combined number of \(\$ 8\) and \(\$ 10\) tickets sold was 7 times the number of \(\$ 12\) tickets sold?

Step-by-Step Solution

Answer

The number of $8 tickets sold is 210, the number of $10 tickets sold is 140, and the number of $12 tickets sold is 50.

Step 1: Write down the given equations

Based on the given problem, we can form the equations:\n1. \(x + y + z = 400\) (total number of tickets sold)\n2. \(8x + 10y + 12z = 3700\) (total revenue)\n3. \(x + y = 7z\) (relation between the number of $8 and $10 tickets and the number of $12 tickets)

Step 2: Substitution

From equation 3, we get \(z = (x + y) / 7\). Substitute z in equations 1 and 2:\n1. \(x + y + (x + y) / 7 = 400\)\n2. \(8x + 10y + 12*(x + y) / 7 = 3700\)

Step 3: Solve the equations

Now you can solve equations in step 2 using any method of solving simultaneous equations to get the values of x, y and z. Thus solving these will give x = 210, y = 140 and z = 50.

Key Concepts

Algebraic Equations

Algebraic equations are the cornerstone of algebra, representing relationships between variables and constants. They are statements of equality that involve algebraic expressions on both sides of the equal sign. For instance, in the context of the college production of Evita, we can translate quantities, such as the number of tickets sold and the total income, into algebraic expressions and set up equations to find the values we're seeking.

In our example, we have three types of tickets, each with a different price, contributing to the total income. By establishing variables for the number of each type of ticket sold (\(x, y, z\)), we're able to create the following algebraic equations:

• Total tickets sold: \(x + y + z = 400\)
• Total income: \(8x + 10y + 12z = 3700\)
• Relation between ticket types: \(x + y = 7z\)

System of Equations

A system of equations is a set of two or more equations with the same set of variables. Solving these systems means finding values for the variables that make all the equations true simultaneously. There are several methods to solve systems, such as substitution, elimination, and graphical analysis.

In our ticket problem, we constructed a system of equations to represent the scenario. To solve it effectively, we used the method of substitution, which involves expressing one variable in terms of others and then replacing that variable in the other equations. This consolidation helps reduce the complexity of the system, making it easier to solve for each unknown sequentially.

• Substitution of \(z\): \(z = (x + y) / 7\)
• Simplified system:
  • \(x + y + (x + y) / 7 = 400\)
  • \(8x + 10y + 12*(x + y) / 7 = 3700\)

Problem-Solving in Algebra

Problem-solving in algebra involves translating word problems into mathematical statements and then using various algebraic methods to find solutions. The core of problem-solving is comprehension of the problem, identification of what is unknown, and the creation of a strategy to solve for those unknowns.

With the Evita production problem, we first understood the scenario and then outlined our unknowns: the number of tickets of each price sold. Next, we formulated a strategy by setting up a system of equations and chose substitution as our method for solving the system. Upon substitution, we simplified and solved the equations to find the number of each type of ticket sold, completing the problem-solving process. This approach not only aids in finding the solution but also fosters a deeper understanding of the underlying mathematical principles involved.