Problem 42 - Algebra A Combined Function

Question

California 1 South is a historic highway that stretches 123 miles along the coast from Monterey to Morro Bay. Suppose that two cars start driving this highway, one from each town. They meet after 3 hours. Find the rate of each car if one car travels 1 mile per hour faster than the other car.

Step-by-Step Solution

Answer

The slower car travels at 20 mph, and the faster car travels at 21 mph.

Step 1: Define the Variables

Let's define the variables first: let the speed of the slower car be \( x \) miles per hour. Then, the speed of the faster car, which is 1 mile per hour faster, will be \( x + 1 \) miles per hour.

Step 2: Setup the Equation

Since both cars meet after 3 hours and they together cover the total distance of 123 miles, we can express this as an equation: \( 3x + 3(x+1) = 123 \).

Step 3: Simplify the Expression

Let's simplify the expression: \[3x + 3x + 3 = 123,\] combine like terms: \[6x + 3 = 123.\]

Step 4: Solve for x

First, subtract 3 from both sides: \[6x = 120.\] Next, divide by 6 to solve for \( x \): \[x = 20.\] This means the slower car was traveling at 20 miles per hour.

Step 5: Find the Rate of the Faster Car

Since the faster car is traveling 1 mile per hour more than the slower car, its speed is \( x + 1 = 21 \) miles per hour.

Key Concepts

Linear Equations

Linear equations are fundamental in algebra as they express relationships between variables in a proportional way. They often appear in the form of \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable. In our highway problem, both cars’ distances over time need to equal the total distance of 123 miles. Hence, we form a linear equation: \( 3x + 3(x+1) = 123 \). Here, both terms on the left side represent the distances each car travels in three hours. Breaking down complex problems into linear equations helps in simplifying the problem, making it easier to find the solution.

• Linear Equations express relationships with constant rates
• The equation \( 3x + 3(x+1) = 123 \) represents the combined distances covered by the two cars
• They simplify problem-solving by creating a straightforward relationship

Variable Definition

Defining variables is crucial in solving problems mathematically. A variable is a symbol that stands in for an unknown quantity or a changing value. In the problem, we first assign a variable to represent a known unknown: the speed of the slower car. Let’s denote this as \( x \).
Since the faster car travels 1 mile per hour more, its speed is represented as \( x + 1 \).
This strategic setup helps us translate a real-world situation into a mathematical model, where each variable corresponds to a specific part of the problem.

• Variables act as placeholders for unknowns
• In our scenario, \( x \) represents the speed of the slower car
• Choosing appropriate variables simplifies further solution processes

Equation Solving

Solving equations involves a series of manipulations that balance both sides to isolate and find the value of the unknown variable. In our problem, we begin with the equation \( 3x + 3(x+1) = 123 \). Simplifying this requires:

• First, expanding the equation
• Combining like terms to get \( 6x + 3 = 123 \)
• Isolating \( x \) by subtracting 3, resulting in \( 6x = 120 \)
• Finally, dividing by 6 to yield \( x = 20 \)

These steps, if followed precisely, lead us to the desired solution, in this case, the speed of the slower car being 20 miles per hour.

Problem Solving Steps

Effective problem solving in algebra often follows a structured approach, which can be replicated across various problems. The process is as follows:
First, clearly define the variables to represent unknowns (step 1). Next, set up the equation using the relationship defined by the problem (step 2). Simplify the expression by performing basic algebraic operations (step 3). Solve for the variable by isolating it through steps of subtraction and division (step 4). Finally, calculate any additional values needed (step 5).

• Defining variables sets the foundation for solving
• Constructing the equation models the problem mathematically
• Simplifying lays the groundwork for easy calculation
• Each step brings us closer to the solution

Using these steps systematically not only helps in finding the correct answer but also builds strong problem-solving skills in algebra.