Math Problem Statement

Use the​ Gauss-Jordan elimination method to find all solutions of the system of linear equations. left brace Start 2 By 1 Matrix 1st Row 1st Column 2nd Row 1st Column EndMatrix

4 x minus 4 y equals 4 negative 12 x plus 12 y equals negative 12 Question content area bottom Part 1 Write the augmented matrix corresponding to the system. Select the correct choice below​ and, if​ necessary, fill in the answer boxes to complete your choice. A.left bracket Start 3 By 2 Table 1st Row 1st Column 4 2nd Column nothing 2nd Row 1st Column negative 4 2nd Column nothing 3rd Row 1st Column 4 2nd Column nothing EndTable right bracket left bracket Start 3 By 2 Table 1st Row 1st Column 4 2nd Column nothing 2nd Row 1st Column negative 4 2nd Column nothing 3rd Row 1st Column 4 2nd Column nothing EndTable right bracket         

B.left bracket Start 2 By 4 Table 1st Row 1st Column 4 2nd Column negative 4 3rd Column 4 4st Column 1 2nd Row 1st Column nothing 2nd Column nothing 3rd Column nothing 4st Column nothing EndTable right bracket left bracket Start 2 By 4 Table 1st Row 1st Column 4 2nd Column negative 4 3rd Column 4 4st Column 1 2nd Row 1st Column nothing 2nd Column nothing 3rd Column nothing 4st Column nothing EndTable right bracket            

C.left bracket Start 2 By 3 Table 1st Row 1st Column 4 2nd Column negative 4 3rd Column 4 2nd Row 1st Column nothing 2nd Column nothing 3rd Column nothing EndTable right bracket left bracket Start 2 By 3 Table 1st Row 1st Column 4 2nd Column negative 4 3rd Column 4 2nd Row 1st Column nothing 2nd Column nothing 3rd Column nothing EndTable right bracket

Solution

Let's start by writing the system of linear equations in augmented matrix form. The given system of equations is:

4x - 4y &= 4 \quad \text{(Equation 1)} \\ -12x + 12y &= -12 \quad \text{(Equation 2)} \end{aligned}$$ The augmented matrix corresponding to this system is formed by writing the coefficients of the variables $$x$$ and $$y$$ in the first two columns, and the constants on the right side of the equations in the last column. The matrix should look like this: $$\begin{pmatrix} 4 & -4 & 4 \\ -12 & 12 & -12 \end{pmatrix}$$ So the correct choice is: **C.** $$\left[ \begin{array}{ccc} 4 & -4 & 4 \\ -12 & 12 & -12 \end{array} \right]$$ Would you like to proceed with solving the system using the Gauss-Jordan elimination method? ### Related Questions: 1. How do you perform row operations on a matrix? 2. What are the possible outcomes of solving a system of linear equations using Gauss-Jordan elimination? 3. Can a system of linear equations have infinitely many solutions, and how can it be identified? 4. How do you interpret the results of the augmented matrix after applying the Gauss-Jordan method? 5. What is the difference between Gauss-Jordan elimination and Gaussian elimination? **Tip:** In Gauss-Jordan elimination, the goal is to transform the matrix into reduced row-echelon form, where each leading entry in a row is 1 and is the only non-zero entry in its column.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Matrix Operations

Formulas

Augmented matrix form
Gauss-Jordan elimination method

Theorems

Gauss-Jordan elimination theorem

Suitable Grade Level

Grades 10-12

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