Problem 36 Solve each system using the Gaus... [FREE SOLUTION]

Chapter 4: Problem 36

Solve each system using the Gauss-Jordan elimination method. $$ \begin{array}{r} x+y=3 \\ y=6 \end{array} $$

Short Answer

Expert verified

The solution is $ x = -3 $ and $ y = 6 $.

Step by step solution

Write the system of equations in augmented matrix form

The given system of equations is: 1. $ x + y = 3 $2. $ y = 6 $Write this system in augmented matrix form:\[\begin{pmatrix}1 & 1 & | & 3 \0 & 1 & | & 6\end{pmatrix}\]

Apply row operations to get the matrix in row-echelon form

The matrix is already in row-echelon form, where the leading coefficients (pivots) are 1.

Make the elements below and above each pivot to be 0 (Gauss-Jordan form)

Subtract row 2 from row 1:\[R1 \rightarrow R1 - R2 \begin{pmatrix}1 & 1 & | & 3 \0 & 1 & | & 6\end{pmatrix}\rightarrow\begin{pmatrix}1 & 0 & | & -3 \0 & 1 & | & 6\end{pmatrix}\]

Write the resulting equations from the final matrix

Convert the final matrix back to the system of equations:\[\begin{cases} x = -3 \ y = 6 \end{cases} \]

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

System of Linear Equations

A system of linear equations consists of multiple linear equations working together. Each equation represents a line in a multi-dimensional space. The solution to the system is any point where all the lines intersect.
For example, given the system: $$ \begin{array}{r} x + y = 3 \ y = 6 \end{array} $$, each equation represents a line, and finding the common point where these lines intersect gives us the solution.

The equation $ x + y = 3 $ describes a line where the sum of $ x $ and $ y $ is always 3.
The equation $ y = 6 $ dictates that $ y $ is always 6, no matter the value of $ x $.

When solving these, our goal is to find values for $ x $ and $ y $ that satisfy all equations simultaneously.

Augmented Matrix

An augmented matrix is a compact way to represent a system of linear equations. It combines the coefficients and constants of the equations into a matrix format, which makes it easier to manipulate and solve.

For the given system:

$ x + y = 3 $
$ y = 6 $

We can write the augmented matrix as: \[ \begin{pmatrix}1 & 1 & | & 3 \ 0 & 1 & | & 6 \end{pmatrix} \]Here:

The first row $ \begin{pmatrix}1 & 1 & | & 3 \end{pmatrix} $ represents the equation $ x + y = 3 $.
The second row $ \begin{pmatrix}0 & 1 & | & 6 \end{pmatrix} $ stands for $ y = 6 $.

The vertical line separates the coefficients on the left from the constants on the right.

Row Operations

Row operations are techniques used to simplify matrices, making it easier to solve the system of equations. They involve:

Swapping two rows
Multiplying a row by a non-zero number
Adding or subtracting the multiple of one row to another

For our example,

Firstly, the matrix $ \begin{pmatrix}1 & 1 & | & 3 \ 0 & 1 & | & 6 \end{pmatrix} $ is already in an easy form.
Using row operations, we want to make the elements below and above each pivot (leading entry in each column) zero.
We subtract row 2 from row 1: $ \text{R1} \rightarrow \text{R1} - \text{R2} $.

Now we can directly read the solutions from the matrix.

Row-Echelon Form

Row-echelon form (REF) is a special form of a matrix where:

Each leading entry (pivot) of a row is 1.
Each leading entry is to the right of the leading entry in the row above.
The rows with all zeros, if any, are at the bottom.

For our given matrix: \[ \begin{pmatrix}1 & 1 & | & 3 \ 0 & 1 & | & 6 \end{pmatrix} \]

It is already in REF since the pivot positions are in the correct order and have leading 1s.

However, to solve the system using Gauss-Jordan elimination, we aim for a more precise form called reduced row-echelon form (RREF), where the pivot positions are isolated, with zeros above and below.

After applying necessary row operations, our final matrix becomes: \[ \begin{pmatrix}1 & 0 & | & -3 \ 0 & 1 & | & 6 \end{pmatrix} \]which is in RREF.

This directly tells us the values of $ x $ and $ y $, which are $ x = -3 $ and $ y = 6 $, respectively.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Solve each system using the Gauss-Jordan elimination method. $$ \begin{array}{r} x+y=3 \\ y=6 \end{array} $$

Short Answer

Step by step solution

Write the system of equations in augmented matrix form

Apply row operations to get the matrix in row-echelon form

Make the elements below and above each pivot to be 0 (Gauss-Jordan form)

Write the resulting equations from the final matrix

Key Concepts

System of Linear Equations

Augmented Matrix

Row Operations

Row-Echelon Form

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Applied Mathematics

Decision Maths

Geometry

Discrete Mathematics

Mechanics Maths

Study anywhere. Anytime. Across all devices.