Textbook Question
In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists.
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 21
For Exercises 11–22, use Cramer's Rule to solve each system.
Verified step by step guidance1
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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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 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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Introduction to Systems of Linear Equations
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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Related Practice
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. -5(A+D)
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Textbook Question
Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
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Textbook Question
Let and . 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
Let and . Solve each matrix equation for X. 3X + 2A = B