Problem 100 - Algebra and Trigonometry

Question

The toll to a bridge costs \(8.00. Commuters who frequently use the bridge have the option of purchasing a monthly discount pass for \)36.00. With the discount pass, the toll is reduced to $5.00. For how many bridge crossings per month will the cost without the discount pass be the same as the cost with pass? What will be the monthly cost for each option?

Step-by-Step Solution

Answer

The costs will be the same if the bridge is crossed 12 times in a month. The monthly cost for each option is $96.

Step 1: Define the Costs

The cost without the discount pass, which is a fixed rate, can be represented by the formula \( cost_{without} = 8x \), where \( x \) is the number of times the bridge is crossed. The cost with the discount pass includes a fixed fee plus the discounted rate. This can be represented by the formula \( cost_{with} = 36 + 5x \).

Step 2: Set the Costs Equal

Next, set the two cost expressions to be equal and solve for \( x \). Hence, \( 8x = 36 + 5x \).

Step 3: Solve for x

To solve for \( x \), first subtract \( 5x \) from both sides to get \( 3x = 36 \). Then divide both sides by 3 to isolate \( x \), which gives \( x = 12 \).

Step 4: Calculate Total Costs

Finally, calculate the cost for each scenario. For the cost without the discount pass, multiply the number of crossings (12) by the regular fee (8), which gives \( cost_{without} = 8*12 = 96 \) dollars. For the cost with the discount pass, add the cost of the pass (36) to the product of the number of crossings (12) and the discount fee (5), which gives \( cost_{with} = 36+5*12 = 96 \) dollars.

Key Concepts

Linear Equations

Linear equations are fundamental to algebra, representing relationships between variables with straight lines when plotted on a graph. Essentially, they form the equation of a line, typically expressed in the form of \( y = mx + b \), where \( m \) denotes the slope of the line and \( b \) represents the y-intercept, the point where the line crosses the y-axis.

In our bridge toll problem, we deal with linear equations that relate the total monthly cost to the number of times the bridge is crossed. Through these equations, you can visualize the costs as straight lines, with each crossing being a move along the x-axis (number of crossings), which affects the cost on the y-axis (total cost).

Moreover, the equations also reflect two different scenarios – crossing with and without a discount pass. This makes it easy to compare the options and understand how the number of crossings affects each option differently. As such, you gain insight into the problem's linear nature, which leads to finding a solution at the break-even point where these lines intersect, indicating equal cost for either option.

Solving Linear Equations

Solving linear equations is the process of finding the value of the variable that makes the equation true. These variables, which are often represented by letters such as \( x \) or \( y \), play a vital role in establishing the relationship described by the equation.

For instance, in the context of the bridge toll scenario, solving the linear equation involves finding the number of crossings that equates both costs. Following the step-by-step solution, you simplify the equations and isolate the variable \( x \) to determine its value. This process typically involves combining like terms and using inverse operations, such as addition and subtraction to cancel out terms, or multiplication and division to solve for the variable.

Overall, the ability to manipulate these equations is crucial, as it allows us to navigate through various real-life problems that can be expressed algebraically. By practicing with clear examples like this, you enhance your skills in algebra and can apply them in other contexts with confidence.

Algebraic Representation

Algebraic representation refers to using algebraic expressions and equations to model real-world scenarios. These mathematical constructs translate the problem's details into a form that can be analyzed and solved using algebraic methods.

In our problem, the costs for using the bridge with and without a monthly discount pass are represented algebraically. The cost without the pass is a direct proportion of the number of crossings, symbolized by \( cost_{without} = 8x \). The cost with the pass has two components: a fixed monthly pass fee and a reduced toll, collectively represented by \( cost_{with} = 36 + 5x \).

By creating these algebraic expressions, not only do we capture the essence of the toll system, but we also construct a bridge (pun intended) to the world of algebra, where we can leverage the power of equations to find answers to our problem. Such representations are invaluable in various fields, from economics to engineering, showcasing the universality of algebra in solving everyday issues.