Skip to main content
Ch. 6 - Matrices and Determinants
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 6 - Matrices and DeterminantsProblema 33
Capitolo 7, Problema 33

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
{2x+2y+7z=−12x+y+2z=24x+6y+z=15\(\begin{cases}\) 2x + 2y + 7z = -1 \\ 2x + y + 2z = 2 \\ 4x + 6y + z = 15 \(\end{cases}\)

Guida verificata passo dopo passo
1
Write the system of equations as an augmented matrix. For the system: \[\begin{cases} 2x + 2y + 7z = -1 \\ 2x + y + 2z = 2 \\ 4x + 6y + z = 15 \end{cases}\] The augmented matrix is: \[\left[ \begin{array}{ccc|c} 2 & 2 & 7 & -1 \\ 2 & 1 & 2 & 2 \\ 4 & 6 & 1 & 15 \end{array} \right]\]
Use Gaussian elimination to transform the matrix into an upper triangular form. Start by using the first row to eliminate the \(x\)-terms in the second and third rows. For example, subtract the first row from the second row, and subtract twice the first row from the third row.
Continue the elimination process to get zeros below the pivot in the second column. Use the second row to eliminate the \(y\)-term in the third row by appropriate row operations.
Once the matrix is in upper triangular form, write the corresponding system of equations from the matrix. This system will have the form: \[\begin{cases} a_{11}x + a_{12}y + a_{13}z = b_1 \\ 0 + a_{22}y + a_{23}z = b_2 \\ 0 + 0 + a_{33}z = b_3 \end{cases}\]
Use back-substitution to solve for \(z\) from the third equation, then substitute \(z\) into the second equation to solve for \(y\), and finally substitute \(y\) and \(z\) into the first equation to solve for \(x\).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
7m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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 matrix methods.
Video consigliato:
4:27
Introduction to Systems of Linear Equations

Matrix Representation of Systems

Systems of linear equations can be expressed in matrix form as AX = B, where A is the coefficient matrix, X is the column matrix of variables, and B is the constants matrix. This representation simplifies the use of matrix operations to solve the system efficiently.
Video consigliato:
6:19
Systems of Inequalities

Gaussian Elimination and Gauss-Jordan Elimination

Gaussian elimination transforms the augmented matrix into an upper triangular form to solve via back-substitution, while Gauss-Jordan elimination reduces it further to reduced row-echelon form for direct solution. Both methods use row operations to systematically solve linear systems.
Video consigliato:
6:48
Solving Systems of Equations - Elimination
Pratica correlata
Domanda del libro di testo

In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate each determinant. ∣1561451910∣\(\begin{vmatrix}\) 1 & 5 & 6 \\ 1 & 4 & 5 \\ 1 & 9 & 10 \(\end{vmatrix}\)

775
views
Domanda del libro di testo

Write each matrix equation as a system of linear equations without matrices.

[20−1030110][xyz]=[695]\(\begin{bmatrix}\) 2 & 0 & -1 \\ 0 & 3 & 0 \\ 1 & 1 & 0 \(\end{bmatrix}\) \(\begin{bmatrix}\) x \\ y \\ z \(\end{bmatrix}\) = \(\begin{bmatrix}\) 6 \\ 9 \\ 5 \(\end{bmatrix}\)

587
views
Domanda del libro di testo

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

{w+x+y+z=42w+x−2y−z=0w−2x−y−2z=−23w+2x+y+3z=4\(\begin{cases}\) w + x + y + z = 4 \\ 2w + x - 2y - z = 0 \\ w - 2x - y - 2z = -2 \\ 3w + 2x + y + 3z = 4 \(\end{cases}\)

926
views
Domanda del libro di testo

In Exercises 27 - 36, find (if possible) the following matrices: a. AB b. BA 1 - 1 4 1 1 0 A = 4 - 1 3 B = 1 2 4 2 0 - 2 1 - 1 3

142
views
Domanda del libro di testo

In Exercises 27 - 36, find (if possible) the following matrices: a. AB b. BA 4 2 2 3 4 A = 6 1 B = 3 5 - 1 - 2 0

920
views
Domanda del libro di testo

Write each matrix equation as a system of linear equations without matrices.

[4−72−3][xy]=[−31]\(\begin{bmatrix}\) 4 & -7 \\ 2 & -3 \(\end{bmatrix}\) \(\begin{bmatrix}\) x \\ y \(\end{bmatrix}\) = \(\begin{bmatrix}\) -3 \\ 1 \(\end{bmatrix}\)

624
views