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Ch. 6 - Matrices and Determinants
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 6 - Matrices and DeterminantsProblem 7
Chapter 7, Problem 7

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. {8x+5y+11z=30−x−4y+2z=32x−y+5z=12\(\begin{cases}\)8x + 5y + 11z = 30 \\-x - 4y + 2z = 3 \\2x - y + 5z = 12\(\end{cases}\)

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1
Write the system of equations as an augmented matrix: \[\begin{bmatrix} 8 & 5 & 11 & | & 30 \\ -1 & -4 & 2 & | & 3 \\ 2 & -1 & 5 & | & 12 \end{bmatrix}\]
Use row operations to create a leading 1 in the first row, first column if possible, or proceed to eliminate the x-terms in the second and third rows by adding suitable multiples of the first row to those rows.
Perform row operations to eliminate the x-term from the second and third rows, aiming to get zeros below the leading coefficient in the first column.
Next, focus on the second row to create a leading 1 in the y-position, then use it to eliminate the y-term in the third row.
Finally, use back substitution starting from the last row to express variables in terms of constants or parameters, thereby finding the complete solution or determining if no solution exists.

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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 the 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 through 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

Write the augmented matrix for each system of linear equations.

{2w+5x−3y+z=23x+y=4w−x+5y=95w−5x−2y=1\(\begin{cases}\)2w + 5x - 3y + z = 2 \\3x + y = 4 \(\w\) - x + 5y = 9 \\5w - 5x - 2y = 1\(\end{cases}\)

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Textbook Question

In Exercises 5 - 8, find values for the variables so that the matrices in each exercise are equal. [x2yz9]=[41239]\(\begin{bmatrix}\)x & 2y \(\z\) & 9\(\end{bmatrix}\)=\(\begin{bmatrix}\)4 & 12 \\3 & 9\(\end{bmatrix}\)

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Textbook Question

Write the system of linear equations represented by the augmented matrix. Use x, y, and z, or, if necessary, w, x, y, and z, for the variables.

[503∣−1101−4∣12720∣3]\(\begin{bmatrix}\)5 & 0 & 3 & \(\vert\) & -11 \\0 & 1 & -4 & \(\vert\) & 12 \\7 & 2 & 0 & \(\vert\) & 3\(\end{bmatrix}\)

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Textbook Question

In Exercises 8–11, use Gaussian elimination to find the complete solution to each system, or show that none exists.

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Textbook Question

Find the products AB and BA to determine whether B is the multiplicative inverse of A.

A=[010001100],B=[001100010]A = \(\begin{bmatrix}\) 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \(\end{bmatrix}\), \(\quad\) B = \(\begin{bmatrix}\) 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \(\end{bmatrix}\)

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Textbook Question

Evaluate each determinant in Exercises 1–10.

∣−5−1−2−7∣\(\begin{vmatrix}\)-5 & -1 \\-2 & -7\(\end{vmatrix}\)

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