Problem 12 - Algebra 2

Question

If \(y\) varies inversely as \(x\) and \(y=5\) when \(x=10,\) find \(y\) when \(x=2\)

Step-by-Step Solution

Answer

When \(x=2\), \(y=25\).

Step 1: Understanding Inverse Variation

In inverse variation, two variables are related so that their product is always constant. If \(y\) varies inversely as \(x\), the relationship can be expressed as \(y = \frac{k}{x}\), where \(k\) is the constant of variation.

Step 2: Finding the Constant of Variation

We are given that \(y = 5\) when \(x = 10\). Substitute these values into the inverse variation formula: \(5 = \frac{k}{10}\). Solve for \(k\) by multiplying both sides by 10 to get \(k = 50\).

Step 3: Applying the Constant to a New Situation

Now, use the constant of variation \(k = 50\) to find \(y\) when \(x = 2\). Substitute \(x = 2\) into the equation \(y = \frac{50}{x}\) to get \(y = \frac{50}{2}\).

Step 4: Calculating the New Value of y

Simplify the expression \(y = \frac{50}{2}\) to find \(y = 25\). This means that when \(x = 2\), \(y = 25\).

Key Concepts

Constant of Variation

In inverse variation, the constant of variation, often symbolized as \( k \), plays a crucial role in understanding how two variables are interrelated. Think of it as the glue holding the relationship between \( y \) and \( x \) in a consistent pattern. Once the relationship is established, this constant remains unchanged regardless of the values of the variables.To find the constant of variation in inverse variation, use the formula:

• \( y = \frac{k}{x} \)

Here, when \( y \) is 5 and \( x \) is 10, substituting these into the formula gives us:

• \( 5 = \frac{k}{10} \)

Solving this equation, multiply both sides by 10 to isolate \( k \):

• \( k = 50 \)

This value of \( k \) is what you'll use to find other values of \( y \) when \( x \) changes.

Relationship Between Variables

Understanding the relationship between variables in an inverse variation is essential. In this context, as one variable increases, the other decreases. This relationship can be a bit counter-intuitive, especially when compared to direct variation, where both variables increase or decrease together.In inverse variation, the equation \( y = \frac{k}{x} \) shows:

• \( y \) decreases as \( x \) increases
• \( y \) increases as \( x \) decreases

This characteristic features a hyperbolic graph. It's important to note that although \( y \) and \( x \) do not alter simultaneously in the same direction, their product always equals the constant \( k \).This inverse relationship holds for any pair of numbers, ensuring that \( y \times x = k \) remains true no matter what values \( y \) and \( x \) take on.

Solve Inverse Problems

Solving inverse variation problems involves using the constant of variation to find unknown values. Once you have determined \( k \), it serves as a foundation for future calculations, enabling you to solve various inverse variation scenarios efficiently.For example, in the given problem, with \( k = 50 \), you can quickly find \( y \) for a new \( x \) value by substituting into the inverse formula:

• For \( x = 2 \), substitute into \( y = \frac{50}{x} \)
• This calculation becomes \( y = \frac{50}{2} = 25 \)

Using \( k \), you are assured of consistency regardless of the values you plug in.The beauty of inverse problems lies in their straightforward nature once the constant is known. Each new scenario builds confidence as the relationship remains predictable. Whether in academic exercises or real-world applications, understanding this problem-solving approach can be very beneficial.