Textbook Question
Write the augmented matrix for each system of linear equations.
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Ch. 6 - Matrices and Determinants
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 7, Problem 5
In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists.
Verified step by step guidance1
Write the system of equations as an augmented matrix:
\[\left[\begin{array}{ccc|c} 3 & 4 & 2 & 3 \\ 4 & -2 & -8 & -4 \\ 1 & 1 & -1 & 3 \end{array}\right]\]
Use row operations to create a leading 1 in the first row, first column. For example, swap rows if needed or divide the first row by the coefficient of \(x\) in the first equation.
Eliminate the \(x\)-terms from the second and third rows by replacing those rows with suitable linear combinations of the first row and themselves, aiming to get zeros below the leading 1 in the first column.
Move to the second row and create a leading 1 in the second column (pivot position). Then eliminate the \(y\)-term from the third row by using row operations.
Finally, use back substitution starting from the last row to express \(z\), then substitute back to find \(y\), and then \(x\), thus finding the complete solution to the system.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Systems of Linear Equations
A system of linear equations consists of multiple linear equations involving the same set of variables. The goal is to find values for these variables that satisfy all equations simultaneously. Understanding how to represent and interpret these systems is fundamental before applying solution methods.
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Introduction to Systems of Linear Equations
Gaussian Elimination
Gaussian elimination is a systematic method for solving systems of linear equations by transforming the system's augmented matrix into row-echelon form using row operations. This process simplifies the system, making it easier to solve by back substitution or to determine if no solution exists.
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Row Operations and Consistency
Row operations include swapping rows, multiplying a row by a nonzero scalar, and adding multiples of one row to another. These operations preserve the solution set and help identify if the system is consistent (has at least one solution) or inconsistent (no solution). Recognizing inconsistent rows is key to concluding no solution exists.
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Related Practice
Textbook Question
Evaluate each determinant in Exercises 1–10.
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Textbook Question
In Exercises 5 - 8, find values for the variables so that the matrices in each exercise are equal.
Textbook Question
Write the augmented matrix for each system of linear equations.
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Textbook Question
In Exercises 5 - 8, find values for the variables so that the matrices in each exercise are equal.
Textbook Question
Find the products AB and BA to determine whether B is the multiplicative inverse of A.
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