Select All The Correct Answers.For The First 80 Miles Of Her Road Trip, Lina Travels 10 Miles/hour Slower Than She Did For The Remaining 50 Miles Of Her Trip. If { S $}$ Represents Lina's Speed During The First Part Of Her Trip, Which

Introduction

In this article, we will delve into a mathematical problem that requires us to think critically and apply our knowledge of algebra to solve. The problem revolves around Lina's road trip, where she travels at different speeds for the first 80 miles and the remaining 50 miles. We will use algebraic equations to represent the situation and solve for the unknown speed.

Understanding the Problem

Lina travels 10 miles/hour slower during the first 80 miles of her trip compared to the remaining 50 miles. If we represent her speed during the first part of the trip as { s $}$, we need to find the value of { s $}$. To do this, we will set up an equation that represents the situation and solve for the unknown speed.

Setting Up the Equation

Let's assume that Lina's speed during the first 80 miles is { s $}$ miles/hour. Since she travels 10 miles/hour slower during this part of the trip, her speed for the remaining 50 miles is { s + 10 $}$ miles/hour. We can set up an equation based on the fact that the total time taken for the entire trip is the same.

The time taken to travel the first 80 miles is { \frac{80}{s} $}$, and the time taken to travel the remaining 50 miles is { \frac{50}{s + 10} $}$. Since the total time is the same, we can set up the equation:

$$\frac{80}{s} = \frac{50}{s + 10}$$

Solving the Equation

To solve the equation, we can start by cross-multiplying:

$$80(s + 10) = 50s$$

Expanding the left-hand side of the equation, we get:

$$80s + 800 = 50s$$

Subtracting { 50s $}$ from both sides of the equation, we get:

$$30s = -800$$

Dividing both sides of the equation by { 30 $}$, we get:

$$s = -\frac{800}{30}$$

Simplifying the fraction, we get:

$$s = -\frac{80}{3}$$

Interpreting the Result

The value of { s $}$ represents Lina's speed during the first 80 miles of her trip. However, since speed cannot be negative, we need to re-examine our solution. Let's go back to the equation and try to solve it again.

Alternative Solution

Let's try to solve the equation by factoring:

$$80(s + 10) = 50s$$

Expanding the left-hand side of the equation, we get:

$$80s + 800 = 50s$$

Subtracting { 50s $}$ from both sides of the equation, we get:

$$30s = -800$$

Dividing both sides of the equation by { 30 $}$, we get:

$$s = -\frac{800}{30}$$

However, this solution is still negative. Let's try to solve the equation by using a different approach.

Using a Different Approach

Let's assume that Lina's speed during the first 80 miles is { s $}$ miles/hour. Since she travels 10 miles/hour slower during this part of the trip, her speed for the remaining 50 miles is { s - 10 $}$ miles/hour. We can set up an equation based on the fact that the total time taken for the entire trip is the same.

The time taken to travel the first 80 miles is { \frac{80}{s} $}$, and the time taken to travel the remaining 50 miles is { \frac{50}{s - 10} $}$. Since the total time is the same, we can set up the equation:

$$\frac{80}{s} = \frac{50}{s - 10}$$

Solving the Equation

To solve the equation, we can start by cross-multiplying:

$$80(s - 10) = 50s$$

Expanding the left-hand side of the equation, we get:

$$80s - 800 = 50s$$

Adding { 800 $}$ to both sides of the equation, we get:

$$80s = 50s + 800$$

Subtracting { 50s $}$ from both sides of the equation, we get:

$$30s = 800$$

Dividing both sides of the equation by { 30 $}$, we get:

$$s = \frac{800}{30}$$

Simplifying the fraction, we get:

$$s = \frac{80}{3}$$

Conclusion

In this article, we solved a mathematical problem that required us to think critically and apply our knowledge of algebra to solve. We set up an equation that represented the situation and solved for the unknown speed. We also used a different approach to solve the equation and found that the value of { s $}$ represents Lina's speed during the first 80 miles of her trip.

Final Answer

The final answer is { \frac{80}{3} $}$.