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Inverse variation is when one quantity increases as another quantity decreases. The relationship between these two quantities can be expressed mathematically. If you multiply the two quantities together, you get a constant value. This idea is crucial for solving problems where you need to find how changes in one variable affect another.
In the exercise, the number of cavities Lori develops varies inversely with the time she spends brushing her teeth. This means the product of the number of cavities and the brushing time remains constant. Mathematically, if we denote the number of cavities as y and the brushing time as x, the formula is: \( xy = k \), where k is the constant.

The constant of variation is a crucial element when dealing with inverse variation problems. It represents the constant product of the related variables. In our exercise, we first need to identify the variables and plug in their values to find this constant.
In the problem, Lori brushes her teeth for 0.5 minutes and has 4 cavities. Substituting these values into the equation \( x \times y = k \), we get:
\( 0.5 \times 4 = k \)
Thus, the constant of variation, k, is 2.
With this constant, we can predict the number of cavities for different brushing times.

Solving inverse variation problems involves a series of well-defined steps:

• Identify the variables: Determine which quantities vary inversely. In this case, the number of cavities (y) and the brushing time (x).
• Find the constant of variation: Use the given values to calculate the constant k using the equation \( xy = k \).
• Use the inverse variation equation: With the constant known, you can find the unknown variable for different given values.

In Lori's case, we first identified the variables as brushing time and cavities. Then, we used given values to find the constant, and finally substituted the brushing time of 2 minutes into the equation to predict the number of cavities.

Elementary algebra is the foundation for many mathematical concepts and problem-solving techniques. Understanding how inverse variation fits into algebra will help you tackle many real-world problems.
In elementary algebra, you will often manipulate equations to find unknowns. For instance, in inverse variation problems, you solve for the constant first and then use it to find unknown values by rearranging the equation. In our exercise, these steps involve solving a simple equation and understanding the relationships between variables, such as:
If \( x = 2 \) and \( k = 2 \), solve for y by rearranging the equation \( xy = k \) to \( y = \frac{k}{x} \).
So, understanding the basic algebraic principles of isolating variables and solving equations is crucial for solving inverse variation problems effectively.