Questions: Write an augmented matrix and use elementary row operations in order to solve the following system of equations. Your final matrix should be in reduced row echelon form. In order to get credit you will have to have a correct final answer as accurate steps in each row operation. (x - 5y = 7) (5x + 8y = -4) Write the augmented matrix: [ ][ ][ ] [ ][ ][ ]

Solution Steps

To solve the given system of equations using matrices, we first write the system as an augmented matrix. Then, we perform elementary row operations to transform the matrix into reduced row echelon form (RREF). This will allow us to read off the solutions for \(x\) and \(y\).

The given system of equations is: \[ \begin{align_} x - 5y &= 7 \\ 5x + 8y &= -4 \end{align_} \] We can represent this system as an augmented matrix: \[ A = \begin{bmatrix} 1 & -5 & 7 \\ 5 & 8 & -4 \end{bmatrix} \]

We will perform elementary row operations to convert the matrix into reduced row echelon form (RREF).

1. Eliminate the first element of the second row: \[ R_2 = R_2 - 5R_1 \implies R_2 = \begin{bmatrix} 0 & 1 & -1.18181818 \end{bmatrix} \]

2. Normalize the second row: \[ R_2 = \frac{R_2}{1} \implies R_2 = \begin{bmatrix} 0 & 1 & -1.18181818 \end{bmatrix} \]

3. Eliminate the second element of the first row: \[ R_1 = R_1 + 5R_2 \implies R_1 = \begin{bmatrix} 1 & 0 & 1.09090909 \end{bmatrix} \]

The matrix is now in RREF: \[ \begin{bmatrix} 1 & 0 & 1.09090909 \\ 0 & 1 & -1.18181818 \end{bmatrix} \]

From the RREF, we can read the solutions: \[ x = 1.0909 \quad \text{(rounded to four significant digits)} \] \[ y = -1.1818 \quad \text{(rounded to four significant digits)} \]

Final Answer