Problem 42 - Algebra 2

Question

The total cost \(c\) in dollars to go to a fair and ride \(n\) roller coasters is given by the equation $$c=15+3 n$$ If the total cost was \(\$ 33\) , how many roller coasters were ridden? F. 6 G. 7 H. 8 J. 9

Step-by-Step Solution

Answer

The number of roller coasters ridden is 6 (Option F).

Step 1: Understanding the Problem

We are given a linear equation \( c = 15 + 3n \) that represents the total cost \( c \) as a function of the number of roller coasters \( n \). We need to find the value of \( n \) when the total cost \( c \) is \$33.

Step 2: Substitute the Given Total Cost

Substitute \( c = 33 \) into the equation: \[ 33 = 15 + 3n \] This equation represents the relationship between the total cost and the number of roller coasters ridden.

Step 3: Isolate the Term with \( n \)

Subtract 15 from both sides to isolate the term with \( n \): \[ 33 - 15 = 3n \] which simplifies to \[ 18 = 3n \].

Step 4: Solve for \( n \)

Divide both sides by 3 to solve for \( n \): \[ n = \frac{18}{3} \] which results in \[ n = 6 \].

Step 5: Check Calculation

Verify by substituting \( n = 6 \) back into the original equation: \[ c = 15 + 3 \times 6 = 15 + 18 = 33 \]. The total cost is indeed \$33, confirming our solution.

Key Concepts

Solving Equations

Solving equations is a core aspect of algebra that allows us to find the unknown variable in an equation. In the context of the original exercise, we were tasked with determining how many roller coasters were ridden when the total cost was given. This involves solving a linear equation, which typically has the format of \( ax + b = c \). The key steps generally include identifying the equation, substituting known values, and finding the value of the unknown variable.

The process of solving the equation involves isolating the variable on one side of the equation. We applied this by substituting the cost \( c = 33 \) into the equation \( c = 15 + 3n \). Then, we used algebraic techniques to manipulate the equation, resulting in a value for \( n \). By subtracting 15 from both sides and then dividing by the coefficient of \( n \), which is 3, we were able to find the answer, \( n = 6 \).

• Recognize the given equation and variable to solve.
• Substitute known values in the equation.
• Use inverse operations to isolate the variable.

These steps are fundamental skills required when working with any type of algebraic equations.

Algebraic Manipulation

Algebraic manipulation involves using various mathematical techniques to simplify and solve equations. This can include adding, subtracting, multiplying, and dividing both sides of an equation to gradually isolate the variable in question.

In our problem, the equation \( 33 = 15 + 3n \) was a classic example where algebraic manipulation was used efficiently. We started by subtracting 15 from both sides to get \( 18 = 3n \), which simplified the equation substantially. The next step involved dividing each side of the equation by 3. This gave us \( n = 6 \), isolating \( n \) and providing a solution to the problem. These manipulations are essential in simplifying equations to a point where the solution can be seen clearly.

• Use addition or subtraction to eliminate constants from one side.
• Apply division or multiplication to cancel out coefficients.
• Simplify systematic steps to arrive at a solution.

Algebraic manipulation is a tool that strengthens problem-solving skills in mathematics.

Real-world Applications

Linear equations are not just abstract concepts — they have plenty of real-world applications. They are used to describe relationships and can model numerous situations realistically.

In this exercise, we dealt with an equation representing a practical scenario: calculating the total cost of attending a fair and riding roller coasters. In real-life, linear equations like \( c = 15 + 3n \) model predictable costs, profits, or other quantities. Such equations are fundamental in economics, business, engineering, and daily decision-making. For instance, understanding how costs scale with activity can help set budgets or predict expenses.

• Understanding relationships between different real-world entities.
• Predicting changes and outcomes based on linear functions.
• Calculating costs, savings, or efficiencies in various situations.

Applying linear equations to real-world scenarios helps in making informed decisions and forecasts.