Problem 243 - Elementary Algebra

Question

In the following exercises, translate to a system of equations and solve. Tickets for a dance recital cost \(\$ 15\) for adults and \(\$ 7\) for children. The dance company sold 253 tickets and the total receipts were \(\$ 2,771\). How many adult tickets and how many child tickets were sold?

Step-by-Step Solution

Answer

125 adult tickets and 128 child tickets were sold.

Step 1: Identify Variables

Let the number of adult tickets be denoted by \(A\) and the number of child tickets be denoted by \(C\).

Step 2: Write the System of Equations

From the problem, we have two pieces of information that will form our equations:1. The total number of tickets sold is 253: \(A + C = 253\).2. The total receipts from the tickets are \$2,771: \(15A + 7C = 2771\).

Step 3: Solve the First Equation for One Variable

Solve the first equation for \(A\):\[A = 253 - C\]

Step 4: Substitute into the Second Equation

Substitute \(A = 253 - C\) into the second equation:\[15(253 - C) + 7C = 2771\]

Step 5: Simplify and Solve for C

Distribute and combine like terms:\[3795 - 15C + 7C = 2771\]\[3795 - 8C = 2771\]Isolate \(C\):\[8C = 1024\]\[C = 128\]

Step 6: Solve for A

Substitute \(C = 128\) back into \(A = 253 - C\):\[A = 253 - 128\]\[A = 125\]

Key Concepts

linear equations

Linear equations are equations in which the highest power of the variable is one. They graph as straight lines and can often represent real-world situations, like the ticket problem in our exercise. To solve linear equations, you will often deal with two types:

• Single-variable linear equations: These are simple equations where you solve for one variable, like \(x + 5 = 10\).
• Systems of linear equations: These involve two or more linear equations working together. In our problem, we need to find both adult and child tickets, so we use a system of equations.

These equations help us understand relationships and make predictions. For example, knowing how ticket sales relate to revenue helps the dance company plan future events.

substitution method

The substitution method is a way to solve systems of equations by replacing one variable with an expression involving the other variable. It involves the following steps:
1. Solve one of the equations for one variable. For example, we solved \(A + C = 253\) for \(A\), giving us \(A = 253 - C\).
2. Substitute (replace) this expression into the other equation. We substituted \(A = 253 - C\) into \(15A + 7C = 2771\), resulting in a single equation with one variable, \(15(253 - C) + 7C = 2771\).
3. Solve the simplified equation for the remaining variable. In our case, we found \(C = 128\).
4. Use the solution from step 3 to find the second variable by substituting back. Finally, substituting \(C = 128\) back into \(A = 253 - C\) gives \(A = 125\).
The substitution method is especially useful when one equation is easily solvable for one variable, making it a straightforward yet powerful tool for solving systems of equations.

real-world application

Solving systems of equations has many real-world applications. In our example, it helps determine the number of adult and child tickets sold at a dance recital based on total sales and receipts. Here are some other ways this concept is used:

• Budgeting: Businesses and individuals use these equations to allocate funds and predict expenses.
• Resource management: It's useful in managing resources like staffing, materials, and time in various projects.
• Scientific research: Scientists use systems of equations to model physical phenomena and solve complex problems.

In everyday life, from planning events to running businesses, being able to set up and solve systems of equations is a crucial skill. It allows you to make informed decisions and effectively manage resources.