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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 21
Capitolo 7, Problema 21

For Exercises 11–22, use Cramer's Rule to solve each system. {2x=3y+25x=51−4y\(\begin{cases}\) 2x = 3y + 2 \\ 5x = 51 - 4y \(\end{cases}\)

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Rewrite both equations in standard form (Ax + By = C). For the first equation, subtract 3y and 2 from both sides to get: \(2x - 3y = 2\).
For the second equation, subtract 51 and add 4y to both sides to get: \(5x + 4y = 51\).
Identify the coefficients for the system: \(A = \begin{bmatrix} 2 & -3 \\ 5 & 4 \end{bmatrix}\) and the constants vector \(\mathbf{C} = \begin{bmatrix} 2 \\ 51 \end{bmatrix}\).
Calculate the determinant of matrix \(A\), denoted as \(D\), using the formula \(D = a_{11}a_{22} - a_{12}a_{21}\), where \(a_{ij}\) are the elements of matrix \(A\).
Find determinants \(D_x\) and \(D_y\) by replacing the respective columns of \(A\) with the constants vector \(\mathbf{C}\), then solve for \(x\) and \(y\) using Cramer's Rule: \(x = \frac{D_x}{D}\) and \(y = \frac{D_y}{D}\).

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Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. The goal is to find values for the variables that satisfy all equations simultaneously. Understanding how to represent and manipulate these systems is fundamental for solving them.
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Cramer's Rule

Cramer's Rule is a method for solving systems of linear equations using determinants. It applies when the system has the same number of equations as variables and the determinant of the coefficient matrix is non-zero. Solutions are found by replacing columns of the coefficient matrix with the constants vector and calculating determinants.
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Determinants of 2x2 Matrices

The determinant of a 2x2 matrix [[a, b], [c, d]] is calculated as ad - bc. This scalar value helps determine if the system has a unique solution (non-zero determinant) and is essential in applying Cramer's Rule to solve for variables.
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Determinants of 2×2 Matrices
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In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. {w+x−y+z=−22w−x+2y−z=7−w+2x+y+2z=−1\(\begin{cases}\) w + x - y + z = -2 \\ 2w - x + 2y - z = 7 \\ -w + 2x + y + 2z = -1 \(\end{cases}\)

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

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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. 3X + 2A = B

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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 \(\end{cases}\)

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