Problem 32 - Algebra 1: Concepts and Skills

Question

use the following information. You can run 200 meters per minute uphill and 250 meters per minute downhill. One day you run a total of 2200 meters in 10 minutes. Assign labels to the verbal model below. Then write an algebraic model.

Step-by-Step Solution

Answer

The algebraic model for this situation is a system of two linear equations: \(x + y = 2200\) and \(x/200 + y/250 = 10\), where \(x\) represents the uphill distance, and \(y\) the downhill distance.

Step 1: Identify and label the variables

Let the uphill distance covered be represented by \(x\), and the downhill distance covered be represented by \(y\). These are the main variables to consider, represented by lengths in meters. It is known that \(x + y = 2200\) (in meters) since the total distance run both uphill and downhill should add up to 2200 meters.

Step 2: Establish rates and time

The rates of motion are given: 200 meters per minute uphill and 250 meters per minute downhill. The total time for the run is 10 minutes. Therefore, the expression for the time running uphill is \(x/200\), and for running downhill is \(y/250\), where \(x\) and \(y\) are expressed in meters, so the units match up with the units of the rates. The total time spent running uphill and downhill should be equal to 10 minutes, so we have \(x/200 + y/250 = 10\).

Step 3: Write the algebraic model

The algebraic model for this situation is represented by a system of two linear equations, comprised of the equation representing the total distance and the equation representing the total time. The system of equations is written as follows: \(x + y = 2200\) and \(x/200 + y/250 = 10\).

Key Concepts

Linear Equations

Linear equations are foundational in algebra and describe relationships using simple, linear expressions. They often take the form of a straight line when graphed. In the context of running distances, they help express conditions such as total distance covered and time taken. An example is given in the exercise, where the total distance condition is represented by the linear equation:

• \( x + y = 2200 \)

This states that the sum of the distances covered running uphill and downhill equals 2200 meters.
Another linear equation represents the time condition, where both uphill and downhill times add to a total of 10 minutes:

• \( \frac{x}{200} + \frac{y}{250} = 10 \)

Together, these two equations form a system of linear equations, which is a set of equations with multiple variables that you solve simultaneously. Solving these equations will reveal the individual distances covered uphill and downhill.

Variables

Variables are symbols that represent unknown values in mathematical expressions and equations. In algebra, they are essential in forming expressions that model real-world scenarios. In our running problem, we have two main variables:

• \( x \) : representing the uphill distance covered in meters
• \( y \) : representing the downhill distance covered in meters

Using variables allows us to set up equations that describe the situation concisely. By defining \( x \) and \( y \), we can clearly and systematically work through the problem.
This systematic approach helps you manipulate and solve the linear equations by substituting and rearranging terms, ultimately finding the values of \( x \) and \( y \). The concept of variables is central to developing an algebraic model, as they serve as placeholders for the values we need to discover.

Distance-Time Relationship

The distance-time relationship is a fundamental concept in understanding motion. It helps link how far objects travel with the time taken, influenced by their speed. The formula for calculating this is

• Distance = Speed × Time

In our scenario, the speeds uphill and downhill are different, making it crucial to consider each separately:
When running uphill:
- Speed is 200 meters per minute.
- Thus, time running uphill is represented as \( \frac{x}{200} \).When running downhill:
- Speed is 250 meters per minute.
- Hence, time running downhill is \( \frac{y}{250} \).The combined distance-time equation, as derived in the exercise, is crucial to solve for the actual distances covered uphill and downhill. The total time spent on the run is always 10 minutes, regardless of how the distances individually break down.