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Chapter 3: Problem 42

If \(4 x=2 y=z\) and \(x y z=64, x\) is: (A) 16 (B) 4 (C) 8 (D) 6 (E) 2

Short Answer

Expert verified
x is 2 (E)

Step by step solution

01

- Express variables in terms of x

Given the equations, first express y and z in terms of x. From the first equation, we have: \[4x = 2y = z\] Therefore, \( y = 2x \) and \( z = 4x \).
02

- Substitute and solve for x

Next, substitute \( y = 2x \) and \( z = 4x \) into the second equation: \[ x y z = 64 \]\[ x (2x) (4x) = 64 \] This simplifies to: \[ 8x^3 = 64 \] Divide both sides by 8: \[ x^3 = 8 \] Taking the cube root of both sides results in: \[ x = 2 \]

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Key Concepts

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

algebra
Algebra is a foundational branch of mathematics that focuses on working with variables and constants. In many problems, like the one in our exercise, you'll need to manipulate equations to find the value of an unknown. When solving algebraic equations, keep these tips in mind:
  • Simplify equations where possible
  • Use inverse operations to isolate the variable
  • Substitute values as needed to make solving easier
Here, we started by expressing variables in terms of another to make the substitution process smoother.
variable substitution
Variable substitution is a powerful technique in algebra that involves replacing one variable with another expression or value. This is particularly useful when dealing with multiple equations. In our example, the given relationships are:

\(4x = 2y = z\)

From this, we can rewrite y and z in terms of x:

\(y = 2x\)
\(z = 4x\)

Substituting these back into the main equation helps to simplify the problem. This method is key in breaking down complex problems and making them more manageable.
cubic equations
Cubic equations are polynomial equations of the form \(ax^3 + bx^2 + cx + d = 0\). In our problem, after substitution, we got:

\(8x^3 = 64\)

We simplified this by dividing both sides by 8 to get:

\(x^3 = 8\)

To solve for x, we took the cube root of both sides:

\(x = 2\)

Recognizing and solving cubic equations often involves isolating the cubic term and taking the appropriate root. Practicing this technique will enhance your algebraic problem-solving skills.
GMAT preparation
The GMAT is a critical test for many business school applicants, and mastering its math section is essential. Here are some tips for tackling GMAT math problems:
  • Understand the underlying concepts, such as algebra, geometry, and arithmetic
  • Practice variable substitution to simplify complex problems
  • Familiarize yourself with recognizing and solving cubic and quadratic equations
Consistent practice and understanding these core concepts will improve your problem-solving speed and accuracy. Make sure to review past questions and solutions to understand the different question types you'll encounter.

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