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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 27
Chapter 6, Problem 27

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with z arbitrary.
2x - y = 6
4x - 2y = 0

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1
Write the system of equations as an augmented matrix. For the system \( 2x - y = 6 \) and \( 4x - 2y = 0 \), the augmented matrix is: \[ \left[\begin{array}{cc|c} 2 & -1 & 6 \\ 4 & -2 & 0 \end{array}\right] \]
Use row operations to get a leading 1 in the first row, first column. You can divide the first row by 2: \[ R_1 \rightarrow \frac{1}{2} R_1 \] which gives: \[ \left[\begin{array}{cc|c} 1 & -\frac{1}{2} & 3 \\ 4 & -2 & 0 \end{array}\right] \]
Eliminate the first variable (x) from the second row by replacing the second row with \( R_2 - 4R_1 \): \[ R_2 \rightarrow R_2 - 4R_1 \] This will create a zero in the first column of the second row.
Simplify the second row after the operation to see if it leads to a row of zeros or a contradiction. This will help determine if the system has a unique solution, no solution, or infinitely many solutions.
If the system has infinitely many solutions, express one variable (in this case \( y \)) as arbitrary and write the other variable(s) in terms of this arbitrary variable. If there is a unique solution, back-substitute to find the values of \( x \) and \( y \).

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Key Concepts

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

Gauss-Jordan Elimination Method

Gauss-Jordan elimination is a systematic procedure to solve systems of linear equations by transforming the augmented matrix into reduced row-echelon form. This method uses row operations to simplify the matrix until each variable corresponds to a leading 1, making the solution straightforward to read.
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Types of Solutions in Linear Systems

A system of linear equations can have a unique solution, infinitely many solutions, or no solution. Infinitely many solutions occur when equations are dependent, leading to free variables that can take arbitrary values, which must be expressed explicitly in the solution.
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Expressing Solutions with Arbitrary Variables

When a system has infinitely many solutions, one or more variables are free and can be assigned arbitrary parameters (like y or z). Writing the solution set involves expressing dependent variables in terms of these arbitrary variables to describe all possible solutions.
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Related Practice
Textbook Question

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.

x2+2y2=9 x^2 + 2y^2 = 9

x2+y2=25x^2+y^2=25

Textbook Question

Graph each inequality. y ≤ log(x - 1) - 2

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Textbook Question

Solve each system, using the method indicated.

3x + y = -7

x - y = -5 (Gaussian elimination)

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Textbook Question

Graph each inequality. y > 2x + 1

Textbook Question

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

(2x-1)/3 + (y+2)/4 = 4

(x+3)/2 - (x-y)/2 = 3

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Textbook Question

Find the partial fraction decomposition for each rational expression. See Examples 1–4. 1/(x(2x + 1)(3x2 + 4))