Problem 5 - Intermediate Algebra

Question

Fill in the blanks. Elementary ______ operations can be used on an augmented matrix to produce a simpler equivalent matrix that gives the solution of a system. This process is called ____ - _____ elimination.

Step-by-Step Solution

Answer

The blanks are 'row' and 'Gaussian elimination'.

Step 1: Identify the type of operations

In linear algebra, the term 'elementary operations' refers to operations used on matrices and equations to simplify and solve them. These are called elementary row operations. They include row switching, row multiplication, and row addition.

Step 2: Understand what process uses these operations

The elementary row operations are typically used in the process of transforming a matrix to solve systems of linear equations. This method is well-known as Gaussian elimination.

Step 3: Connect the steps to fill in the blanks

With these understandings in mind, the first blank should be filled with 'row', indicating the type of elementary operations. The second and third blanks should be completed with 'Gaussian' and 'elimination', respectively, since the process is known as Gaussian elimination.

Key Concepts

Augmented Matrix

An augmented matrix is a very useful tool when dealing with systems of linear equations. It combines the coefficients of the variables and the constants from the equations into a single, easy-to-handle matrix.
This makes the process of finding solutions more straightforward. To create an augmented matrix, you align the coefficients of each variable from the left until you reach the constants on the right.

• If you have a system of equations: \[\begin{align*} 2x + y &= 5, \ 4x - 3y &= -2 \end{align*}\] The augmented matrix would look like: \[ \begin{bmatrix} 2 & 1 & | & 5 \ 4 & -3 & | & -2 \end{bmatrix} \]

This format allows for easy application of row operations and is essential for applying the Gaussian elimination method.

Gaussian Elimination

Gaussian elimination is a method used to solve systems of linear equations. It involves transforming an augmented matrix into a simpler form, usually an upper triangular matrix. This is achieved by performing a series of elementary row operations. The goal is to make the system as simple as possible, ideally achieving an identity matrix on the left side of the augmented matrix.

• Start by selecting a leading (or pivot) element from the first column of the augmented matrix.
• Use row operations to make all the other entries in that column zero.
• Move to the next column and repeat this process until the matrix is in an upper triangular form.

This systematic approach results in working with simpler equations which can be easily solved through back substitution. Understanding Gaussian elimination is fundamental to solving complex systems of equations efficiently.

Systems of Linear Equations

A system of linear equations is a collection of two or more equations with the same set of variables. The goal is to find a common solution set, meaning values that satisfy all the equations simultaneously. Linear equations are equations of the first degree, meaning they do not have variables that are multiplied together or raised to a power other than one.

• An example of a simple system is: \[\begin{align*} x + 2y &= 5, \ 3x - y &= 4 \end{align*}\]
• The solution to this system, if one exists, is the point at which these equations intersect when graphed.
• You can solve systems of linear equations using various methods, such as substitution, elimination, or matrix methods like Gaussian elimination.

By representing these systems as augmented matrices, it becomes much easier to manipulate and solve them using elementary row operations. This provides a clear and structured path to finding solutions.