Problem 10 - Algebra 2

Question

Write a system of equations for each augmented matrix. $$ \left[\begin{array}{rr|r}{-1} & {2} & {-6} \\ {1} & {1} & {7}\end{array}\right] $$

Step-by-Step Solution

Answer

The system of equations is \( -1x + 2y = -6 \) and \( x + y = 7 \).

Step 1: Identify the Elements of the Matrix

Each row of the matrix represents the coefficients and constant term of a linear equation. The left section (before the vertical line) contains the coefficients of the variables, while the right section (after the vertical line) contains the constants.

Step 2: Write the First Equation

From the first row of the matrix \(\begin{array}{rr|r}{-1} & {2} & {-6}\end{array}\), construct the first equation using the coefficients of the variables. The equation is \( -1x + 2y = -6 \) where \( x \) and \( y \) are the variables.

Step 3: Write the Second Equation

From the second row of the matrix \(\begin{array}{rr|r}{1} & {1} & {7}\end{array}\), construct the second equation using the coefficients of the variables. The equation is \( 1x + 1y = 7 \) or simply \( x + y = 7 \).

Key Concepts

Augmented Matrix

Understanding how to work with an augmented matrix is fundamental in the study of linear algebra, especially when solving systems of linear equations. Simply put, an augmented matrix is a combination of the coefficient matrix from a system of linear equations and a column of constant terms.

An augmented matrix looks like this: \[\left[\begin{array}{cc|c} a & b & e \ c & d & f \end{array}\right]\]where the left section holds the coefficients (a, b, c, d) of the variables, and the right section after the vertical line contains the constants (e, f). In the context of our specific exercise, the augmented matrix \[\left[\begin{array}{rr|r}{-1} & {2} & {-6} \ {1} & {1} & {7}\end{array}\right]\]encapsulates a system of linear equations.To improve clarity when dealing with augmented matrices, remember to:

• Clearly separate the coefficient portion from the constants with a vertical line.
• Ensure that each row corresponds to a specific equation from the system.
• Understand that each column correlates to a particular variable across all equations.

This structured format provides an organized approach to solving systems of linear equations using various methods, such as Gaussian elimination or row reduction techniques.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, matrices, and systems of linear equations. It provides a framework for analyzing and solving problems where relationships between quantities are linear, meaning they can be represented as straight lines when graphed.
Central to linear algebra is the concept of a matrix, which is a rectangular array holding numbers or functions that can be manipulated according to specific rules. The study of linear algebra covers matrix operations, determinants, vector spaces, eigenvalues, and more, and has applications in fields as diverse as engineering, physics, computer science, and economics.

To master linear algebra concepts:

• Practice operations with vectors and matrices.
• Learn the different techniques for solving systems of linear equations.
• Explore the geometric interpretations of linear transformations.

Being comfortable with linear algebra is imperative for anyone working in scientific or quantitative fields, as it provides the tools for modeling and solving complex problems.

Solving Systems of Equations

Solving systems of equations is a crucial technique in algebra where you find the values of variables that satisfy multiple equations at once. There are several methods to solve these systems, including substitution, elimination, graphical methods, and using matrices.

When solving systems with matrices, the augmented matrix serves as a bridge to simplified forms that are easier to interpret. For example, you can use row reduction to bring the matrix to row-echelon form or even reduced row-echelon form, which can then be used to easily back-solve for the variable values.To effectively solve systems of equations:

• Be methodical in your approach, applying steps consistently.
• Check your solutions by substituting them back into the original equations.
• Utilize graphing to visually verify the solutions.

Understanding different solving techniques will increase your versatility and problem-solving skills across various types of linear systems.