Problem 48 - Algebra 1

Question

The cost of a taxi ride is an initial fee plus \(\$ 1.50\) for each mile. Your fare for 9 miles is \(\$ 15.50 .\) Write an equation that models the total cost \(y\) of a taxi ride in terms of the number of miles \(x\).

Step-by-Step Solution

Answer

The equation that models the total cost \(y\) of a taxi ride in terms of the number of miles \(x\) is \(y = 1.50x + 2\)

Step 1: Understand the given data

First, interpret the given data. A taxi ride costs $1.50 per mile after a certain initial fee. For a 9-mile ride, the total cost is $15.50, including the initial fee.

Step 2: Form an initial equation

Assume the initial fee to be \(b\). The equation then is \(y = 1.50 \times x + b\) (This equation represents the total taxi fare (y) in terms of number of miles (x), where the $1.50 is the rate per mile and b is the initial flat fee.) Then substitute the given values into the equation to solve for \(b\), i.e., \(15.50 = 1.50 \times 9 + b\).

Step 3: Solve the equation

Simplify the equation to find the initial rate (b). To do this, perform the multiplication on the right side of the equation and then isolate b by subtracting the result from both sides. The simplified equation becomes: \(15.50 = 13.50 + b\). Now, subtract 13.5 from both sides, \(b=15.50 - 13.50\). After performing this subtraction, \(b= 2\) .

Step 4: Form a Final Equation

Substitute the initial fee \(b\) into the original equation, which is \(y = 1.50x + b\). So, the final equation which models the total cost \(y\) of a taxi ride in terms of the number of miles \(x\) will be \(y = 1.50x + 2\)

Key Concepts

Slope-Intercept Form

The slope-intercept form is a popular way to express linear equations. In this format, an equation is typically written as \( y = mx + b \). Here, \( y \) represents the dependent variable, \( x \) is the independent variable, \( m \) is the slope, and \( b \) represents the y-intercept.

• **Why Use Slope-Intercept Form?** It makes it easy to identify the slope and the y-intercept, which are crucial for graphing linear equations.
• **Slope:** In our taxi fare example, the slope is \( 1.50 \). This represents the cost per mile. In mathematical terms, the slope \( m \) is the rate of change.
• **Y-Intercept:** The initial fee, \( b = 2 \), is our y-intercept. This is the cost when the number of miles \( x \) is zero.

This form facilitates comprehension of how variables interact. It provides insights into both the fixed base cost and variable costs based on miles traveled.

Word Problems

Word problems require translating real-life situations into mathematical equations. They help develop the critical skill of identifying and working with the variables involved in a problem.

• **Understanding the Problem:** Look for clues in the text. For instance, the problem states a base fee plus a rate per mile, which clues us to look for a linear relationship.
• **Assign Variables:** In the exercise, we assigned \( y \) to the total cost and \( x \) to the number of miles. The rate per mile is \( 1.50 \), and \( b \) is the initial cost.
• **Translating to Equations:** The key to solving word problems is creating equations that accurately reflect the scenario. Here, \( y = 1.50x + b \) was formed based on the language of the problem.

Successfully solving word problems allows for applying math skills to various real-world applications. It bridges the gap between theoretical math and practical situations like computing taxi fares.

Solving Equations

Solving equations involves finding values for unknowns that make the equation true. This process often includes simplifying expressions and isolating variables.*Step-by-Step Approach:*

• **Initiate the Equation:** Begin by establishing the relationship. In the taxi problem, we started with \( y = 1.50x + b \).
• **Substitute Known Values:** Insert the given values to solve for unknown variables. After substituting the ride's cost, \( 15.50 \), and miles driven, \( 9 \), we formed an equation: \( 15.50 = 1.50 \times 9 + b \).
• **Simplify the Equation:** Perform arithmetic (multiplying and subtracting) to simplify the equation. Here, calculate \( 1.50 \times 9 = 13.50 \) and solve for \( b \), \( b = 15.50 - 13.50 \).
• **Isolate Variables:** Rearrange to find \( b \), yielding \( b = 2 \).

Once solved, you incorporate the value back into the initial equation, ensuring your final equation accurately models the situation. Practicing these steps leads to mastery in solving various types of equations.