Problem 49 - Elementary Algebra

Question

A patient with heart disease needs to lose 96 pounds. The physician's assistant recommends an average loss of 3 pounds per week. Find the number of weeks needec to lose the weight. Use negative numbers to represent weight loss.

Step-by-Step Solution

Answer

32 weeks

Step 1: Identify the Given Values

The total weight loss required is 96 pounds, and the recommended average loss per week is 3 pounds. We'll represent the weight loss as a negative number: -3 pounds per week.

Step 2: Set Up the Equation

Let the number of weeks be represented by the variable \( w \). The equation needed to solve for \( w \) is \[ -3w = -96 \]

Step 3: Solve for the Number of Weeks

Divide both sides by -3 to isolate \( w \): \[ w = \frac{-96}{-3} \] \[ w = 32 \]

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero. They are often represented with a minus sign, such as -1, -2, or -3. In real-life situations, negative numbers can denote loss, decrease, or opposition.
In the context of weight loss, we use negative numbers to denote a reduction in weight. For example, if a person loses 3 pounds per week, we can express this as -3 pounds per week. Negative numbers are essential when solving problems that involve decreases or reverse directions.
Understanding negative numbers is crucial because you will encounter them regularly, especially in contexts like temperature changes, financial losses, or weight loss.

Isolating Variables

Isolating a variable means solving an equation for one specific variable. It involves rearranging the equation such that the variable of interest stands alone on one side of the equation. This allows us to directly find the value of the variable.
In our exercise, the equation is \(-3w = -96\). Here, we aim to isolate the variable \(w\). To do this, you need to get \(w\) by itself on one side of the equation. We do this by performing the same mathematical operation on both sides of the equation.
Here are the steps to isolate \(w\):

• Identify the variable (in this case, \(w\)) and the coefficient that it's multiplied by.
• Perform inverse operations to both sides of the equation to cancel out these coefficients. For example, dividing both sides by -3 to remove the -3 from \(w\).

Once the variable is isolated, solving the equation becomes straightforward.

Division of Integers

Division of integers involves separating a number into equal parts or groups. When dealing with division and negative numbers, remember the following rules:

• Dividing a positive number by a positive number or a negative number by a negative number results in a positive quotient.
• Dividing a positive number by a negative number or a negative number by a positive number results in a negative quotient.

In our exercise, we need to divide -96 by -3:
We have the equation \(-3w = -96\). To isolate \(w\), divide both sides by -3:
\[ w = \frac{-96}{-3}\]
Since both -96 and -3 are negative, the result of their division is positive:
\[ w = 32 \]
This means it will take 32 weeks to lose 96 pounds with a weight loss of 3 pounds per week. Division of integers helps us determine the precise value of the number of weeks required.