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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 3
Chapter 6, Problem 3

What is the augmented matrix of the following system?
-3x + 5y = 2
6x + 2y = 7

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Identify the coefficients of the variables and the constants from each equation in the system. For the first equation, -3x + 5y = 2, the coefficients are -3 for x and 5 for y, and the constant is 2.
For the second equation, 6x + 2y = 7, the coefficients are 6 for x and 2 for y, and the constant is 7.
Write the coefficients of the variables in a matrix form, placing each equation's coefficients in a separate row. This forms the coefficient matrix:
\[\begin{bmatrix} -3 & 5 \\ 6 & 2 \end{bmatrix}\]
To form the augmented matrix, append the constants as an additional column to the coefficient matrix, resulting in:
\[\begin{bmatrix} -3 & 5 & 2 \\ 6 & 2 & 7 \end{bmatrix}\]

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Key Concepts

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

System of Linear Equations

A system of linear equations 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. Understanding the structure of these equations is essential for representing and solving them using matrices.
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Augmented Matrix

An augmented matrix is a compact representation of a system of linear equations. It combines the coefficient matrix and the constants from the equations into one matrix, where each row corresponds to an equation and the last column contains the constants. This form is useful for applying matrix operations to solve the system.
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Matrix Representation of Equations

Matrix representation involves organizing the coefficients of variables and constants into rows and columns. For example, the coefficients of variables form the main part of the matrix, while the constants form an additional column. This representation simplifies solving systems using methods like Gaussian elimination.
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Related Practice
Textbook Question

What expression in x represents ∣x43x∣\(\left\)| \(\begin{matrix}\) x & 4 \\ 3 & x \(\end{matrix}\) \(\right\)|?

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Textbook Question

Answer each question. By what expression should we multiply each side of (3x - 2)/(x + 4)(3x^2 + 1) = A/(x + 4) + (Bx + C)/(3x^2 + 1) so that there are no fractions in the equation?

Textbook Question

Answer each of the following. When appropriate, fill in the blank to correctly complete the sentence. The following nonlinear system has two solutions, one of which is (___, 3).

2x + y = 1

x2 + y2 = 10

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Textbook Question

Use the substitution or elimination method to solve each system of equations. Identify any inconsistent systems or systems with infinitely many solutions. If a system has infinitely many solutions, write the solution set with y arbitrary.

16x+13y=8\(\frac{1}{6x}\)+\(\frac{1}{3y}\)=8

14x+12y=12\(\frac{1}{4x}\)+\(\frac{1}{2y}\)=12

3
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Textbook Question

Answer each question. What is the product [−574230−166][100010001]\(\left\)[ \(\begin{matrix}\) -5 & 7 & 4 \\2 & 3 & 0 \\-1 & 6 & 6 \(\end{matrix}\) \(\right\)]\(\quad\]\left\)[ \(\begin{matrix}\) 1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \(\end{matrix}\) \(\right\)]?

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Textbook Question

If the dimension of matrix A is 3 ×\(\times\) 2 and the dimension of matrix B is 2 ×\(\times\) 6, then the dimension of AB is ____.