Problem 183 - Elementary Algebra

Question

The ice rink sold 95 tickets for the afternoon skating session, for a total of \(\$ 828 .\) General admission tickets cost \(\$ 10\) each and youth tickets cost \(\$ 8\) each. How many general admission tickets and how many youth tickets were sold?

Step-by-Step Solution

Answer

34 general admission tickets and 61 youth tickets were sold.

Step 1: Define variables

Let \(g\) be the number of general admission tickets and \(y\) be the number of youth tickets sold.

Step 2: Set up equations

We know two things: the total number of tickets sold and the total amount of money collected. Thus, we can create the following equations:\(g + y = 95\) (total number of tickets)\(10g + 8y = 828\) (total money collected)

Step 3: Solve the first equation for one variable

Rewrite the first equation in terms of one variable:\(y = 95 - g\)

Step 4: Substitute into the second equation

Substitute \(y = 95 - g\) into the second equation:\(10g + 8(95 - g) = 828\)

Step 5: Simplify and solve for \(g\)

\(10g + 760 - 8g = 828\)Combine like terms:\(2g + 760 = 828\)Subtract 760 from both sides:\(2g = 68\)Divide by 2:\(g = 34\)

Step 6: Solve for \(y\)

Substitute \(g = 34\) back into the equation \(y = 95 - g\):\(y = 95 - 34\)\(y = 61\)

Step 7: Verify the solution

Check the values in both original equations to verify correctness:\(g + y = 34 + 61 = 95\)\(10g + 8y = 10(34) + 8(61) = 340 + 488 = 828\)

Key Concepts

linear equations

Linear equations form the backbone of algebra. They are equations where the highest power of the variable is one. Understanding linear equations is essential because they appear in various aspects of mathematics and real-world problems.

• In our exercise, we use two linear equations to represent two conditions: the total number of tickets and the total amount of money collected.
• A linear equation can be written in the form of ax + by + c = 0 , where a, b, and c are constants.

In this case, our equations are:

• g + y = 95 (equation 1)
• 10g + 8y = 828 (equation 2)

Linear equations are simple yet powerful tools for describing relationships between variables. They help us find missing information by relating different quantities to each other.

systems of equations

A system of equations is a set of two or more equations with the same variables. Solving a system of equations involves finding values for the variables that make all the equations true simultaneously.

• In our exercise, the system is represented by the two linear equations mentioned earlier.
• To solve this system, we need to find the values of g and y that satisfy both equations at the same time.

One common method of solving systems of equations is by using substitution, as we did in this problem, or by using elimination methods. Understanding systems of equations is crucial because it allows us to solve problems involving multiple constraints or conditions.

variable substitution

Variable substitution is a technique where we replace one variable with an equivalent expression involving another variable. This simplifies the system of equations, making it easier to solve.

• First, we solve one of the equations for a variable. For example, we rewrote the first equation as y = 95 - g .
• Then, we substitute this expression into the other equation to find the value of the other variable.

In our problem, substituting y = 95 - g into the second equation yielded: 10g + 8(95 - g) = 828 .
This simplification helped us to find g = 34 , and subsequently, y = 61 . Substitution is a powerful tool as it reduces the number of variables and equations we need to handle at once.

algebraic problem solving

Algebraic problem solving is the process of finding unknown values by setting up and solving equations. It involves various techniques and critical thinking. Here’s a brief overview of the steps:

• Step 1: Define Variables - We defined g and y for general and youth tickets.
• Step 2: Set up Equations - We used the given information to create equations.
• Step 3: Solve for One Variable - We expressed y in terms of g .
• Step 4: Substitute and Solve - Substituting in the second equation gave us the value of g .
• Step 5: Find the Other Variable - Using the value of g , we found y.
• Step 6: Verify - Finally, we checked our solution to ensure accuracy.

Mastering algebraic problem solving enables students to tackle complex problems systematically and confidently.