Problem 295 - Elementary Algebra

Question

In the following exercises, solve. Two busses leave Billings at the same time. The Seattle bus heads west on I-90 at a speed of 73 miles per hour while the Chicago bus heads east at a speed of 79 miles an hour. How many hours will it take them to be 532 miles apart?

Step-by-Step Solution

Answer

3.5 hours

Step 1: Understand the Problem

Two buses leave from the same place but in opposite directions. One bus goes west at 73 mph and the other goes east at 79 mph. We need to find out how many hours it takes for the two buses to be 532 miles apart.

Step 2: Set Up the Equation

Let the time it takes for the buses to be 532 miles apart be denoted as \( t \) hours. The distance covered by the Seattle bus will be \( 73t \) miles and the distance covered by the Chicago bus will be \( 79t \) miles.

Step 3: Add the Distances

Since the buses are moving in opposite directions, their distances add up. Therefore, the equation is: \[ 73t + 79t = 532 \]

Step 4: Solve the Equation

Combine like terms to form the equation: \[ 152t = 532 \]Solve for \( t \) by dividing both sides by 152: \[ t = \frac{532}{152} \]Simplify the fraction to find the value of \( t \).

Step 5: Calculate the Result

Divide 532 by 152 to get: \[ t = 3.5 \]Thus, it takes 3.5 hours for the buses to be 532 miles apart.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that show the equality of two expressions. They often involve variables, constants, and arithmetic operations. In this context, we use algebraic equations to represent the distance each bus travels over time and set this equal to the total distance between them. By forming an equation, we can solve for the unknown variable representing time.

Relative Speed

Relative speed is the combined speed of two objects moving either towards or away from each other. In this exercise, both buses are moving apart, hence their speeds add up to increase the distance between them faster. The relative speed here is the sum of the individual speeds of the two buses. This concept helps us determine how quickly the gap between them increases. We used:

• Speed of Seattle bus: 73 mph
• Speed of Chicago bus: 79 mph

Their combined speed is 73 + 79 = 152 mph.

Solving Linear Equations

To solve a linear equation, you need to isolate the variable by performing operations that simplify the equation. Here, we start with the equation for total distance:
\( 73t + 79t = 532 \)
Combining like terms gives: \( 152t = 532 \)
To solve for \( t \), divide both sides by 152:
\( t = \frac{532}{152} \) Simplifying the fraction:
\( t = 3.5 \) hours.

Distance Problems

Distance problems involve calculating how far something travels over a period of time, given the speed. The formula to remember is:
\( \text{Distance} = \text{Rate} \times \text{Time} \).
In this case, we have two rates (speeds of the buses) and one time variable. By setting up the correct equation, combining distances, and solving for time, we find out how long it takes for two moving objects to be a certain distance apart.

Elementary Algebra

Elementary algebra includes basic algebraic concepts such as working with variables, arithmetic operations, and simple equations. This exercise uses:

• Variables to represent time and distance
• Addition, multiplication, and division operations

Combining these with real-world contexts helps understand and solve practical problems like the one given. Simplifying and manipulating equations is a core part of learning elementary algebra.