Problem 28 - Algebra 2

Question

If \(y\) varies jointly as \(x\) and \(z\) and \(y=\frac{1}{8}\) when \(x=\frac{1}{2}\) and \(z=3,\) find \(y\) when \(x=6\) and \(z=\frac{1}{3}\).

Step-by-Step Solution

Answer

When \( x=6 \) and \( z=\frac{1}{3} \), \( y=\frac{1}{6} \).

Step 1: Understand the Relationship

Joint variation means that \( y \) is proportional to the product of \( x \) and \( z \). Therefore, the relationship can be expressed as \( y = kxz \), where \( k \) is a constant.

Step 2: Find the Constant of Variation

Use the given values \( y = \frac{1}{8} \), \( x = \frac{1}{2} \), and \( z = 3 \) to substitute into the equation \( y = kxz \). Thus, \( \frac{1}{8} = k \cdot \frac{1}{2} \cdot 3 \). Solve for \( k \): \[ k = \frac{1}{8} \div \left(\frac{1}{2} \times 3\right) = \frac{1}{8} \cdot \frac{2}{3} = \frac{1}{12}. \]

Step 3: Substitute New Values and Solve

Now substitute \( x = 6 \), \( z = \frac{1}{3} \), and \( k = \frac{1}{12} \) into the equation \( y = kxz \). Compute \( y = \frac{1}{12} \cdot 6 \cdot \frac{1}{3} \). Simplify this expression: \[ y = \frac{6}{12} \cdot \frac{1}{3} = \frac{1}{2} \cdot \frac{1}{3} = \frac{1}{6}. \]

Step 4: Verify the Calculations

Review the steps: starting with the relationship \( y = kxz \), we calculated \( k = \frac{1}{12} \). Then, using the new \( x \) and \( z \), we found that \( y = \frac{1}{6} \). The calculations confirm the steps were correctly executed.

Key Concepts

Constant of Variation

When dealing with problems involving joint variation, one of the key elements is the **constant of variation**, often represented by the symbol **\( k \)**. This constant acts as a multiplier that adjusts the proportionality between variables.

In the provided exercise, the relationship between \( y \), \( x \), and \( z \) is given by \( y = kxz \). To uncover this constant, you substitute the known quantities into the formula. With \( y = \frac{1}{8} \), \( x = \frac{1}{2} \), and \( z = 3 \), placing these in the equation helps us solve for \( k \).

This particular task involves dividing the given \( y \) by the product of \( x \) and \( z \), eventually simplifying to \( k=\frac{1}{12} \). This value of \( k \) does not change unless the circumstances of the relationship change, maintaining the balance between the variables across different scenarios.

Proportional Relationships

**Proportional relationships** involve two or more quantities changing at the same relative rate. In the context of joint variation, multiple variables influence a resultant variable, which scales proportionally.

In the problem, \( y \) is proportional to the product of \( x \) and \( z \), meaning as \( x \) and \( z \) shift, \( y \) shifts while maintaining the balance established by the constant of variation \( k \).

Understanding this relationship is crucial in such exercises:

• All measures are linked by the constant \( k \).
• The constant ensures the proportional change is consistent and predictable.
• Proportional relationships simplify predictions and calculations by relating variables through \( k \) directly.

Through these relationships, we solve for unknown variables given certain conditions, ensuring the original proportion remains intact.

Algebraic Expressions

In solving problems involving joint variation, **algebraic expressions** are central. These expressions are composed of variables, constants, and operations that represent a mathematical scenario. Here, \( y = kxz \) serves as a foundation.

Within this expression:

• Each symbol stands for a specific quantity: \( y \) is the outcome, \( x \) and \( z \) are the input variables, and \( k \) is the modifier or constant.
• The expression defines how these elements interact.
• Substituting the known values allows us to solve algebraically for the desired variable, by performing operations according to mathematical rules.

By simplifying expressions through operations like multiplication and division, one can uncover unknowns step by step. This problem-solving technique is powerful in algebra, especially when variables are interdependent, as in joint variation challenges.