Question

Solve each system, using the method indicated. x - z = -3 y + z = 6 2x - 3z = -9 (Gauss-Jordan)

Accepted Answer

Write the system of equations as an augmented matrix. The system is:

$$x - z = -3$$
$$y + z = 6$$
$$2x - 3z = -9$$

The variables are $$x$$, $$y$$, and $$z$$, so the augmented matrix is:

$$\begin{bmatrix} 1 & 0 & -1 & | & -3 \ 0 & 1 & 1 & | & 6 \ 2 & 0 & -3 & | & -9 \end{bmatrix}$$ Use row operations to get a leading 1 in the first row, first column (which is already done), and then eliminate the $$x-$$term from the third row by replacing row 3 with (row 3) - 2*(row 1). This will create a zero in the first column of the third row. Next, focus on the third row to get a leading 1 in the $$z-$$column. You may need to multiply the third row by the reciprocal of the coefficient of $$z in $$that row to achieve this. Use the third row to eliminate the $$z-$$terms from the first and second rows by appropriate row operations, aiming to get zeros above and below the leading 1 in the $$z-$$column. Finally, ensure that the second row has a leading 1 in the $$y-$$column and eliminate any other terms in that column from the other rows. After these steps, the matrix should be in reduced row echelon form, from which you can read off the solutions for $$x$$, $$y$$, and $$z.$$

Solve each system, using the method indicated.
x - z = -3
y + z = 6
2x - 3z = -9 (Gauss-Jordan)

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Key Concepts

Systems of Linear Equations

Gauss-Jordan Elimination Method

Matrix Representation of Systems