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Ch. 5 - Systems and Matrices
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 5 - Systems and MatricesProblem 19
Chapter 6, Problem 19

Write the system of equations associated with each augmented matrix . Do not solve.
Calculator screen displaying a 3x4 augmented matrix with rows: [1 1 0 3], [0 2 1 -4], [1 0 -1 5].

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1
Identify the number of variables in the system by counting the number of columns in the augmented matrix, excluding the last column which represents the constants.
Label each variable, typically as \(x_1, x_2, x_3, \ldots\), corresponding to each column of the coefficient matrix.
Write an equation for each row of the matrix by multiplying each variable by its corresponding coefficient in that row and setting the sum equal to the constant term in the last column.
Express each equation in the form \(a_{i1}x_1 + a_{i2}x_2 + \cdots + a_{in}x_n = b_i\), where \(a_{ij}\) are the coefficients from the matrix and \(b_i\) is the constant from the augmented column.
Repeat this process for all rows to form the complete system of linear equations represented by the augmented matrix.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Augmented Matrix

An augmented matrix represents a system of linear equations by combining the coefficient matrix and the constants into one matrix. Each row corresponds to an equation, and each column corresponds to a variable or the constants. Understanding this helps translate between matrix form and equation form.
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System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. Writing the system means expressing each row of the augmented matrix as an equation, with coefficients multiplying variables equal to the constants.
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Matrix to Equation Conversion

Converting an augmented matrix to a system of equations involves interpreting each row as an equation by assigning variables to columns and equating the linear combination of variables to the constant term. This skill is essential for understanding the relationship between matrices and systems.
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Related Practice
Textbook Question

Find the cofactor of each element in the second row of each matrix.

[12−123−2−141]\(\left\)[ \(\begin{matrix}\) 1 & 2 & -1 \\ 2 & 3 & -2 \\ -1 & 4 & 1 \(\end{matrix}\) \(\right\)]

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Textbook Question

Graph each inequality. x ≤ 3

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Textbook Question

Find the inverse, if it exists, for each matrix. [−1−234]\(\left\)[ \(\begin{matrix}\) -1 & -2 \\3 & 4 \(\end{matrix}\) \(\right\)]

Textbook Question

Find the values of the variables for which each statement is true, if possible.

[05x−13y+241z]=[0w6−130418]\(\left\)[ \(\begin{matrix}\) 0 & 5 & x \\ -1 & 3 & y + 2 \\ 4 & 1 & z \(\end{matrix}\) \(\right\)] = \(\left\)[ \(\begin{matrix}\) 0 & w & 6 \\ -1 & 3 & 0 \\ 4 & 1 & 8 \(\end{matrix}\) \(\right\)]

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Textbook Question

Solve each system by elimination. In systems with fractions, first clear denominators.

4x + y = -23

x - 2y = -17

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Find the partial fraction decomposition for each rational expression. See Examples 1–4. (x2)/(x2 + 2x + 1)

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