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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 19
Chapter 7, Problem 19

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. {x+y−2z=23x−y−6z=−7\(\begin{cases}\)x + y - 2z = 2 \\3x - y - 6z = -7\(\end{cases}\)

Verified step by step guidance
1
Write the system of equations as an augmented matrix: \[\left[\begin{array}{ccc|c} 1 & 1 & -2 & 2 \\ 3 & -1 & -6 & -7 \end{array}\right]\]
Use the first row to eliminate the \(x\)-term in the second row. Multiply the first row by 3 and subtract it from the second row: \[R_2 \rightarrow R_2 - 3R_1\]
Perform the row operation to get the new second row: \[\left[\begin{array}{ccc|c} 1 & 1 & -2 & 2 \\ 0 & -4 & 0 & -13 \end{array}\right]\]
Solve the second equation for \(y\) by dividing the entire second row by the coefficient of \(y\): \[y = \frac{-13}{-4} = \frac{13}{4}\]
Substitute the value of \(y\) back into the first equation to solve for \(x\) in terms of \(z\): \[x + \frac{13}{4} - 2z = 2\] Then isolate \(x\) to express it as \[x = 2 - \frac{13}{4} + 2z\]

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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 two or more 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 by back-substitution or to determine if no solution exists.
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Row Operations and Consistency of Systems

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

Let A=[−3−72−950]A = \(\begin{bmatrix}\)-3 & -7 \\2 & -9 \\5 & 0\(\end{bmatrix}\) and B=[−5−1003−4]B = \(\begin{bmatrix}\)-5 & -1 \\0 & 0 \\3 & -4\(\end{bmatrix}\) Solve each matrix equation for X. X - A = B

Textbook Question

In Exercises 14–27, perform the indicated matrix operations given that and D are defined as follows. If an operation is not defined, state the reason. 3A+2D

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Textbook Question

Let A=[−3−72−950]A = \(\begin{bmatrix}\)-3 & -7 \\2 & -9 \\5 & 0\(\end{bmatrix}\) and B=[−5−1003−4]B = \(\begin{bmatrix}\)-5 & -1 \\0 & 0 \\3 & -4\(\end{bmatrix}\). Solve each matrix equation for X. 2X + A = B

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Textbook Question

In Exercises 19–20, a few steps in the process of simplifying the given matrix to row-echelon form, with 1s down the diagonal from upper left to lower right, and 0s below the 1s, are shown. Fill in the missing numbers in the steps that are shown.

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Textbook Question

Perform each matrix row operation and write the new matrix.

[1−111∣301−2−1∣02034∣115124∣6]−2R1+R3−5R1+R4\(\begin{bmatrix}\)1 & -1 & 1 & 1 & \(\vert\) & 3 \\0 & 1 & -2 & -1 & \(\vert\) & 0 \\2 & 0 & 3 & 4 & \(\vert\) & 11 \\5 & 1 & 2 & 4 & \(\vert\) & 6\(\end{bmatrix}\[\quad\]\begin{array}{l}\)-2R_1 + R_3 \\-5R_1 + R_4\(\end{array}\)

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Textbook Question

For Exercises 11–22, use Cramer's Rule to solve each system. {3x−4y=42x+2y=12\(\begin{cases}\)3x - 4y = 4 \\2x + 2y = 12\(\end{cases}\)