Textbook Question
Write each linear system as a matrix equation in the form AX = B, where A is the coefficient matrix and B is the constant matrix.
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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 29
Write each linear system as a matrix equation in the form AX = B, where A is the coefficient matrix and B is the constant matrix.
Verified step by step guidance1
Identify the coefficients of the variables x and y from each equation to form the coefficient matrix A. For the first equation, the coefficients are 6 and 5, and for the second equation, they are 5 and 4. So, matrix A is: \(\begin{bmatrix} 6 & 5 \\ 5 & 4 \end{bmatrix}\).
Next, write the variables x and y as a column matrix X, which is: \(\begin{bmatrix} x \\ y \end{bmatrix}\).
Then, write the constants from the right side of the equations as the constant matrix B. From the equations, the constants are 13 and 10, so matrix B is: \(\begin{bmatrix} 13 \\ 10 \end{bmatrix}\).
Now, express the system of equations in the matrix form \(AX = B\), where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
Putting it all together, the matrix equation is: \(\begin{bmatrix} 6 & 5 \\ 5 & 4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 13 \\ 10 \end{bmatrix}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Linear Systems of Equations
A linear system 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. In this problem, the system has two equations with two variables, x and y.
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Introduction to Systems of Linear Equations
Matrix Representation of Linear Systems
A linear system can be expressed as a matrix equation AX = B, where A is the coefficient matrix containing the coefficients of the variables, X is the column matrix of variables, and B is the constant matrix. This form simplifies solving and analyzing the system.
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Coefficient and Constant Matrices
The coefficient matrix A is formed by the coefficients of variables from each equation, arranged by rows and columns. The constant matrix B contains the constants from the right side of the equations. For example, A = [[6,5],[5,4]] and B = [[13],[10]] for this system.
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Related Practice
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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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