Math Problem Statement

Use Gauss-Jordan row reduction to solve the given system of equations. (If there is no solution, enter NO SOLUTION. If the system is dependent, express your answer using the parameters x, y, z, and/or w.) x − 10y + z − 6w = 1 7x + 5y + z + 7w = 2 8x − 5y + 2z + w = 3 (x, y, z, w) =

Solution

To solve the given system of equations using Gauss-Jordan elimination, we first need to represent the system as an augmented matrix and perform row reduction. The system of equations is:

x - 10y + z - 6w &= 1 \\ 7x + 5y + z + 7w &= 2 \\ 8x - 5y + 2z + w &= 3 \end{aligned}$$ ### Step 1: Write the augmented matrix The augmented matrix representing this system is: $$\begin{bmatrix} 1 & -10 & 1 & -6 & 1 \\ 7 & 5 & 1 & 7 & 2 \\ 8 & -5 & 2 & 1 & 3 \end{bmatrix}$$ ### Step 2: Apply row operations to get the matrix into reduced row echelon form (RREF) We'll use row operations to get leading 1's in each row and eliminate the other values in the columns of those leading 1's. #### Row 1 operations Row 1 is already set up nicely with a leading 1 in the first column, so we can use it to eliminate the values in the first column of rows 2 and 3. Perform the following row operations: - $$R_2 \to R_2 - 7R_1$$ - $$R_3 \to R_3 - 8R_1$$ This gives us: $$\begin{bmatrix} 1 & -10 & 1 & -6 & 1 \\ 0 & 75 & -6 & 49 & -5 \\ 0 & 75 & -6 & 49 & -5 \end{bmatrix}$$ #### Row 2 operations Notice that rows 2 and 3 are the same, which indicates the system may be dependent. We can eliminate one of these rows (since they are identical). We'll divide $$R_2$$ by 75 to make the leading coefficient 1: $$R_2 \to \frac{1}{75}R_2$$ This gives: $$\begin{bmatrix} 1 & -10 & 1 & -6 & 1 \\ 0 & 1 & -\frac{6}{75} & \frac{49}{75} & -\frac{5}{75} \\ 0 & 75 & -6 & 49 & -5 \end{bmatrix}$$ After canceling $$R_3$$,

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Equations
Gauss-Jordan Elimination

Formulas

Row Operations in Matrix Form
Reduced Row Echelon Form (RREF)

Theorems

Gauss-Jordan Elimination Theorem
Consistency and Dependency of Linear Systems

Suitable Grade Level

Grades 11-12 or College Level

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