/*! 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 What is a system of linear equat... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 42

What is a system of linear equations in three variables?

Short Answer

Expert verified
A system of linear equations in three variables consists of three or more linear equations involving the same three variables. It can be represented in the form of \(a_1x + b_1y + c_1z = d_1\), \(a_2x + b_2y + c_2z = d_2\), \(a_3x + b_3y + c_3z = d_3\). The system may have a unique solution, infinite solutions, or no solution, and it can be solved using methods like substitution, elimination, or matrices.

Step by step solution

01

The Definition

A system of linear equations in three variables is a system that consists of three or more linear equations involving three variables. It can be represented as \[ \begin{cases} a_1x + b_1y + c_1z = d_1 \ a_2x + b_2y + c_2z = d_2 \ a_3x + b_3y + c_3z = d_3 \end{cases} \] where \(a_1\), \(b_1\), \(c_1\), \(d_1\), \(a_2\), \(b_2\), \(c_2\), \(d_2\), \(a_3\), \(b_3\), \(c_3\), \(d_3\) are constants and \(x\), \(y\), \(z\) are variables.
02

Possibilities for Solutions

The system of linear equations can either have one unique solution, infinite many solutions, or no solution. \ 1. It has a unique solution if the three planes represented by the equations intersect at a single point. \ 2. It has infinite solutions if the three planes all coincide, meaning they are all the same plane. \ 3. It has no solution if the three planes do not intersect on a single line or at a single point.
03

Solving the System

The system of linear equations in three variables can be solved through various methods such as substitution, elimination, or matrices. \ The goal is to simplify the system to a stage where it is easy to identify the solutions. The procedure involves eliminating one variable at a time until you are able to solve for one of the variables directly.

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Most popular questions from this chapter

Use the two steps for solving a linear programming problem, given in the box on page \(888,\) to solve the problems. On June \(24,1948,\) the former Soviet Union blocked all land and water routes through East Germany to Berlin. A gigantic airlift was organized using American and British planes to bring food, clothing, and other supplies to the more than 2 million people in West Berlin. The cargo capacity was \(30,000\) cubic feet for an American plane and \(20,000\) cubic feet for a British plane. To break the Soviet blockade, the Western Allies had to maximize cargo capacity but were subject to the following restrictions: \(\cdot\) No more than 44 planes could be used. \(\cdot\) The larger American planes required 16 personnel per flight, double that of the requirement for the British planes. The total number of personnel available could not exceed 512 . \(\cdot\) The cost of an American flight was 9000 and the cost of a British flight was 5000 . Total weekly costs could not exceed $300,000 Find the number of American and British planes that were used to maximize cargo capacity.

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