Problem 287 - Elementary Algebra

Question

In the following exercises, solve. Saul drove his truck 3 hours from Dallas towards Kansas City and stopped at a truck stop to get dinner. At the truck stop he met Erwin, who had driven 4 hours from Kansas City towards Dallas. The distance between Dallas and Kansas City is 542 miles, and Erwin's speed was eight miles per hour slower than Saul's speed. Find the speed of the two truckers.

Step-by-Step Solution

Answer

Saul's speed is 82 mph, and Erwin's speed is 74 mph.

Step 1 - Define Variables

Let Saul's speed be denoted as \(S\) (in miles per hour). Therefore, Erwin's speed will be \(S - 8\) mph since Erwin's speed is 8 mph slower than Saul's.

Step 2 - Write the Distance Formulas

The distance each drove can be written using the formula: \[ \text{Distance} = \text{Speed} \times \text{Time} \].So, the distances are:For Saul: \(3S\) miles.For Erwin: \(4(S - 8)\) miles.

Step 3 - Set up the Equation

The sum of the distances driven by Saul and Erwin equals the total distance between Dallas and Kansas City:\(3S + 4(S - 8) = 542\).

Step 4 - Simplify the Equation

Expand and simplify the equation:\(3S + 4S - 32 = 542\)\(7S - 32 = 542\).

Step 5 - Solve for Saul's Speed

Add 32 to both sides of the equation:\(7S = 574\).Then divide both sides by 7:\(S = 82\).

Step 6 - Find Erwin's Speed

Since Erwin's speed is 8 mph slower than Saul's, subtract 8 from Saul's speed:\(82 - 8 = 74\).

Key Concepts

distance-speed-time relationship

The distance-speed-time relationship is essential for solving many real-world problems, including this exercise. The core idea is that distance traveled (\text{d}) is equal to speed (\text{s}) multiplied by time (\text{t}). This is usually represented by the formula: \[ \text{Distance} = \text{Speed} \times \text{Time} \]
Understanding this relationship is crucial. When you know any two of these variables, you can rearrange the equation to find the third. In our exercise, knowing the speeds of Saul and Erwin and their travel times allows us to calculate the distances they've traveled. Getting comfortable with this relationship will make solving such problems easier.

algebraic equations

Algebraic equations are vital tools for solving problems involving unknown quantities. In our given problem, we set up an equation to represent the combined distances Saul and Erwin drove. Using algebra, we turn words into math.
We defined Saul's speed as S (Saul), and because Erwin's speed is 8 mph slower, it's expressed as \(S - 8 \). From these definitions, we can write equations based on the distance-speed-time relationship:
For Saul: \(3 \times S\)
For Erwin: \(4 \times (S - 8)\)
By combining the distance both traveled, we set up the equation:
\[ 3S + 4(S - 8) = 542 \]. Solving this lets us find the unknown speeds.

problem-solving steps

Following a clear problem-solving process is key to tackling word problems effectively. Here's a breakdown:
1. Understand the problem: Read carefully to know what you are asked to find.
2. Define the variables: Assign letters to the unknown quantities (e.g., S for Saul's speed).
3. Set up equations: Use the given relationships and formulas to write equations.
4. Solve the equations: Use algebraic methods to find the variables' values.
5. Check the answer: Verify if calculated values make sense in the context of the problem.
In the solution, each step aligns with this structured approach, helping to solve Saul and Erwin's speeds correctly.

variables and expressions

Using variables and expressions helps simplify and solve complicated problems. A variable, like S in our exercise, represents an unknown number. Expressions, like \(3S\) and \(4(S-8)\), use these variables to form equations.
When working with variables, it's essential to:

• Define them clearly to avoid confusion
• Use them consistently throughout the problem
• Translate words into mathematical expressions accurately

In our exercise, defining Saul's speed as S and Erwin's speed as \(S - 8\) makes it straightforward to set up and solve the equation for their speeds. Proper use of variables and expressions streamlines the problem-solving process.