Questions: 2. (5 points) Consider the linear system of equations below. x1 + x2 + 2x3 + 7x5 = 20 x2 + 3x5 = 1 x3 + 5x5 = 2 The variables in the system are x1, x2, x3, x4, and x5. (a) Express the system as a 3 x 6 augmented matrix. (b) Reduce the system you constructed in part (a) to RREF. (c) Express the solution set of the linear system in parametric vector form.

Solution Steps

(a) To express the system as a \(3 \times 6\) augmented matrix, we need to write the coefficients of each variable in the equations, including zeros for missing variables, and include the constants on the right side of the equations as the last column.

(b) To reduce the matrix to Row-Reduced Echelon Form (RREF), we will use Gaussian elimination, which involves performing row operations to simplify the matrix.

(c) To express the solution set in parametric vector form, we will interpret the RREF matrix to identify the free and leading variables, and express the solution in terms of the free variables.

The given system of equations is: \[ \begin{aligned} x_{1} + x_{2} + 2x_{3} + 7x_{5} &= 20 \\ x_{2} + 3x_{5} &= 1 \\ x_{3} + 5x_{5} &= 2 \end{aligned} \]

To express this system as an augmented matrix, we include coefficients for all variables \(x_1, x_2, x_3, x_4, x_5\) and the constants on the right side: \[ \begin{bmatrix} 1 & 1 & 2 & 0 & 7 & 20 \\ 0 & 1 & 0 & 0 & 3 & 1 \\ 0 & 0 & 1 & 0 & 5 & 2 \end{bmatrix} \]

Using row operations, we reduce the augmented matrix to Row-Reduced Echelon Form (RREF): \[ \begin{bmatrix} 1 & 0 & 0 & 0 & -6 & 15 \\ 0 & 1 & 0 & 0 & 3 & 1 \\ 0 & 0 & 1 & 0 & 5 & 2 \end{bmatrix} \]

From the RREF matrix, we can express the solution in terms of the free variable \(x_5\). The leading variables are \(x_1, x_2, x_3\), and \(x_4\) is not present in the equations, so it is also a free variable. The solutions are: \[ \begin{aligned} x_1 &= 15 + 6x_5 \\ x_2 &= 1 - 3x_5 \\ x_3 &= 2 - 5x_5 \\ x_4 &= x_4 \quad (\text{free variable}) \\ x_5 &= x_5 \quad (\text{free variable}) \end{aligned} \]

\begin{bmatrix} 15 \\ 1 \\ 2 \\ 0 \\ 0 \end{bmatrix} + x_5 \begin{bmatrix} 6 \\ -3 \\ -5 \\ 0 \\ 1 \end{bmatrix} + x_4 \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{bmatrix} \]

Final Answer