Problem 3 - Algebra 2

Question

If \(y\) varies inversely as \(x\) and \(y=-14\) when \(x=12,\) find \(x\) when \(y=21\)

Step-by-Step Solution

Answer

\(x = -8\)

Step 1: Understand the Inverse Variation

When a variable \( y \) varies inversely as \( x \), it means \( y \cdot x = k \), where \( k \) is a constant. So, our main goal is to first find \( k \).

Step 2: Find the Constant \( k \)

We know that \( y = -14 \) when \( x = 12 \). Substitute \( x = 12 \) and \( y = -14 \) into the inverse variation equation to find \( k \): \(-14 \cdot 12 = k\). So, \( k = -168 \).

Step 3: Set Up the Equation with the New \(y\) Value

Now, we apply the value of \( k = -168 \) to find \( x \) when \( y = 21 \). Plug \( y = 21 \) into the inverse relationship: \( 21 \cdot x = -168 \).

Step 4: Solve for \( x \)

Solve the equation \( 21x = -168 \). Divide both sides by 21 to isolate \( x \): \( x = \frac{-168}{21} \).

Step 5: Simplify the Fraction

Simplifying \( \frac{-168}{21} \) gives \( x = -8 \). After division, you can verify that \( 21 \times -8 = -168 \).

Key Concepts

Understanding the Constant of Variation

In inverse variation problems, the constant of variation is a key concept. It's a number that remains unchanged as the variables change. Here, the relationship between the two variables is expressed through a multiplication equation:

• For inverse variation, the product of the variables is constant.
• The formula is: \( y \cdot x = k \)

This is different from direct variation, where one variable is multiplied by a constant to reach the other. An important step is finding this constant, represented by \( k \). For the given problem, we use known values of \( y = -14 \) and \( x = 12 \) to find:\(-14 \cdot 12 = k = -168 \). Once the constant \( k \) is found, it can be used to find other variable values in the problem.

Solving Equations in Inverse Variation

Solving equations in inverse variation involves finding unknown variable values using the constant of variation. Once \( k \) is determined, substitute any other known value and solve for the unknown:

• First, set up the equation using the known constant: if \( y \cdot x = k \), and \( y \) is known, solve for \( x \).

In the problem, after finding the constant \( -168 \), the equation \( 21 \cdot x = -168 \) is established to solve for \( x \). To find \( x \), divide both sides by 21: \( x = \frac{-168}{21} \). These steps highlight the systematic approach needed to isolate and solve for the unknown variable.

Exploring Proportional Relationships

In mathematics, understanding proportional relationships helps in comprehending inverse variations better. In inverse variations, although the two variables multiply to a constant, they don't increase or decrease together:

• If one variable increases, the other decreases to maintain the proportional constant.
• This is opposite to a direct proportional relationship, where the variables change in tandem.

This exercise exemplifies how, as \( y \) changes from \(-14\) to \(21\), \( x \) adjusts accordingly to keep the product \(-168\). It's a beautiful example of balance in numbers, where mathematical principles like inverse relationships maintain consistent results despite changes.