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Chapter 5: Problem 17

If \(x=z / y, y=z / x\) and \(x /(y / z)=64, x\) is: A. 2 B. 4 C. 8 D. 1 E. 16

Short Answer

Expert verified
The value of x is 8 (C).

Step by step solution

01

- State the Given Equations

The problem provides the following equations: 1. \( x = \frac{z}{y} \) 2. \( y = \frac{z}{x} \) 3. \( x \div \left( \frac{y}{z} \right) = 64 \)
02

- Simplify the Third Equation

Simplify the third equation: \( x \div \left( \frac{y}{z} \right) = 64 \) This is equivalent to: \( x \times \frac{z}{y} = 64 \)
03

- Substitute \( \frac{z}{y} \) with x

Notice from the first given equation \( x = \frac{z}{y} \), so substituting \( \frac{z}{y} \) with x, we get: \( x \times x = 64 \)
04

- Solve for x

Simplify the equation from the previous step: \( x^2 = 64 \) Take the square root of both sides: \( x = \sqrt{64} \) Thus, \( x = 8 \)
05

- Verify Consistency with Other Equations

Verify if \( x = 8 \) satisfies the other given equations. Substitute x: For \( x = \frac{z}{y} \): \( 8 = \frac{z}{y} \) For \( y = \frac{z}{x} \): \( y = \frac{z}{8} \) Since these are consistent (both expressions give valid proportional relationships), the solution is confirmed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Equations
Algebraic equations form the backbone of solving GMAT problems. They consist of symbols and numbers connected by arithmetic operations to represent an equality. Understanding how to manipulate these expressions is critical.
For instance, in the exercise provided, we use three algebraic equations: \( x = \frac{z}{y} \), \( y = \frac{z}{x} \), and \( x \div \left( \frac{y}{z} \right) = 64 \).
Each equation relates the variables in different forms, leading to the final solution.
Executing correct operations, like division or multiplication, brings clarity and guides us to the solution.
Variable Substitution
Changing one variable into another is called variable substitution. This tool simplifies complex problems by reducing the number of variables. Here鈥檚 how we applied this technique to our exercise:
Let's start with the third equation: \( x \div \left( \frac{y}{z} \right) = 64 \).
By recognizing that dividing by a fraction is equivalent to multiplying by its reciprocal, we rewrite the equation as: \( x \times \frac{z}{y} = 64 \).
From the initial condition \( x = \frac{z}{y} \), substitute \( \frac{z}{y} \) with x. Now, the equation becomes \( x \times x = 64 \). We reduce the complexity and make it solvable by substituting variables effectively.
Proportional Relationships
Understanding proportions is key in algebra. Proportional relationships show how one variable changes in relation to another.
In our exercise, we have equations that illustrate these relationships: \( x = \frac{z}{y} \) and \( y = \frac{z}{x} \).
These forms demonstrate direct and inverse proportionality. When one variable increases, the other changes accordingly to maintain balance. Recognize this pattern to unravel problems.
For example, if \( x \) triples while \( y \) remains constant, what happens to \( z \)? By keeping track of proportional changes, we can better understand variable dynamics.
Mathematical Consistency
Consistency in mathematical equations ensures that all parts of the problem agree and validate the solution.
We need to verify this by back-checking our values.
In the last step of our exercise, after finding \( x = 8 \), we substitute it back into the original equations:
For \( x = \frac{z}{y} \): \( 8 = \frac{z}{y} \)
For \( y = \frac{z}{x} \): \( y = \frac{z}{8} \)
Both forms match and maintain proportional relationships, confirming the solution鈥檚 correctness. Remember, always verify your answers for consistency across the given equations.

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