/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 42 If \(4 x=2 y=z\) and \(x y z=64,... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

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: \[ x = 2 \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

If \(y=2 / 3 x+5 / 6 x\) and \(x=1\) then \(y\) is: (A) \(2 / 3\) (B) \(7 / 9\) (C) \(7 / 3\) (D) \(7 / 6\) (E) None of the above

Is \(x\) positive or negative? (1) \(x^3+8=0\) (2) \(x^2-4=0\) A. 1 alone, not 2 alone B. 2 alone, not 1 alone C. 1 and 2 together (need both) D. 1 alone or 2 alone E. 1 and 2 together are not sufficient

Is \(x / 8>y / 12 ?\) (1) \(3 x>2 y\) (2) \(x

If \(\mathrm{yx}<1\), then: (A) Both \(x\) and \(y\) are negative (B) Both \(x\) and \(y\) are positive (C) Only one of the two is negative (D) \(x

If \(y=2 x+2 / x\) and \(x=3\) then \(y\) is: (A) \(10 / 3\) (B) \(18 / 3\) (C) \(8^2 / 3\) (D) \(20 / 3\) (E) None of the above
See all solutions

Recommended explanations on English Textbooks

Lexis and Semantics

Read Explanation

Phonetics

Read Explanation

Textual Analysis

Read Explanation

Syntax

Read Explanation

English Language Study

Read Explanation

Cues and Conventions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.