/*! 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 56 There is twice as much ingredien... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 56

There is twice as much ingredient \(X\) by weight as \(Y\) and \(Z\) combined.

Short Answer

Expert verified
Based on the given information, the relationship between the weights of ingredients X, Y, and Z can be represented by the equation \[x = 2y + 2z\], where x is the weight of ingredient X, y is the weight of ingredient Y, and z is the weight of ingredient Z.

Step by step solution

01

Assign Variables

Let's assign the variables for the weights of the ingredients as follows: - x: weight of ingredient X - y: weight of ingredient Y - z: weight of ingredient Z
02

Write the Equation

From the given information, we can deduce that the weight of ingredient X (x) is twice the combined weight of ingredient Y (y) and ingredient Z (z). Mathematically, this can be represented with the following equation: \[x = 2(y+z)\]
03

Simplify the Equation

To simplify the equation, we can distribute the 2 on the right side of the equation to both y and z: \[x = 2y + 2z\] This is the final equation relating the weights of ingredients X, Y, and Z.

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.

Understanding Linear Equations
Linear equations are mathematical expressions that show the relationship between different variables using coefficients and constants. Here, we explore linear equations through the example of ingredient weights. A linear equation, like the one in our exercise, typically has one or more variables, and when represented graphically, produces a straight line.
  • Our exercise deals with a simple linear equation: \(x = 2(y + z)\).
  • This equation indicates how the weight of one ingredient is related to the sum of others.
  • The goal in each linear equation is to find the value, or values, of the variable(s) that make the equation true.
Linear equations are fundamental in both algebra and real-world applications. Recognizing the components鈥攕uch as coefficients and variables鈥攊s key to understanding how the equation behaves under different conditions.
Importance of Variable Assignment
Variable assignment is crucial when dealing with mathematical problems across different contexts. It simplifies complex relationships by representing them with simple placeholders.
  • In our exercise, assigning \(x\), \(y\), and \(z\) to respective ingredient weights allows us to handle the given information clearly.
  • Once assigned, these variables can help us write the mathematical relationship between them, leading to solutions.
By assigning variables correctly, we translate word problems into mathematical expressions, making them easier to manipulate. This method of breaking down complex problems into simpler components by using variables is a foundational skill in mathematics.
Equation Simplification Techniques
Simplifying equations is a powerful skill that aids problem solving by making equations more manageable. In our exercise, we used simplification to better understand and solve the linear equation.
  • Simplification can involve operations like expanding, combining like terms, or factoring.
  • In our example, the simplification involved distributing the 2 across \( (y + z) \), resulting in the equation \(x = 2y + 2z\).
The benefit of simplification is clearer insight into the relationship between variables, and often a quicker path to a solution. By reducing unnecessary complexity, you focus on what's truly important in solving the equation.

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

Aragorn the Great has boasted to his hordes of followers that many a notorious villain has fallen to his awesome sword: His total of 560 victims consists of evil sorcerers, trolls, and orcs. These he has slain with a total of 620 mighty thrusts of his sword; evil sorcerers and trolls each requiring two thrusts (to the chest) and orcs each requiring one thrust (to the neck). When asked about the number of trolls he has slain, he replies, "I, the mighty Aragorn, despise trolls five times as much as I despise evil sorcerers. Accordingly, five times as many trolls as evil sorcerers have fallen to \(\mathrm{m} \mathrm{y}\) sword!" How many of each type of villain has he slain?

Supply and Demand (from the GRE Economics Test) The demand curve for widgets is given by \(D=85-5 P\), and the supply curve is given by \(S=25+5 P\), where \(P\) is the price of widgets. When the widget market is in equilibrium, what is the quantity of widgets bought and sold?

Use technology to solve the systems of equations. Express all solutions as decimals, rounded to one decimal place. $$ \begin{aligned} &2.1 x+0.7 y-1.4 z=-2.3 \\ &3.5 x-4.2 y-4.9 z=3.3 \\ &1.1 x+2.2 y-3.3 z=-10.2 \end{aligned} $$

What is meant by a pivot? What does pivoting do?

Investments: Utility Stocks In October 2008, Consolidated Edison (ED) cost \(\$ 40\) per share and yielded \(6 \%\) per year in dividends, while Southern Union Company (SUG) cost \(\$ 12\) per share and yielded \(4.5 \%\) per year in dividends. \({ }^{9}\) If you invested a total of \(\$ 27,600\) in these stocks and earned \(\$ 1,602\) in dividends in a year, how many shares of each stock did you purchase? (Assume the dividend rate was unchanged for the year.)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

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.