Question

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with z arbitrary.
x + y - 5z = -18
3x - 3y + z = 6
x + 3y - 2z = -13

Accepted Answer

Write the system of equations as an augmented matrix. For the system

$$x + y - 5z = -18$$
$$3x - 3y + z = 6$$
$$x + 3y - 2z = -13$$,

the augmented matrix is:

$$\begin{bmatrix} 1 & 1 & -5 & | & -18 \ 3 & -3 & 1 & | & 6 \ 1 & 3 & -2 & | & -13 \end{bmatrix}$$ Use row operations to get a leading 1 in the first row, first column (which is already 1 here), and then eliminate the $$x-$$terms in the second and third rows by replacing those rows with suitable combinations. For example, replace row 2 with (row 2) - 3*(row 1), and replace row 3 with (row 3) - (row 1). Next, focus on the second row to get a leading 1 in the second column. You may need to divide the entire second row by the coefficient of $$y in $$that row after elimination. Then use this leading 1 to eliminate the $$y-$$term in the third row by replacing row 3 with (row 3) - (appropriate multiple of row 2). After that, get a leading 1 in the third row, third column (for $$z$$), by dividing the third row by the coefficient of $$z in $$that row. Then use this to eliminate the $$z-$$terms in the first and second rows by appropriate row operations. Once the matrix is in reduced row echelon form (RREF), write the system of equations back from the matrix. If the system has a unique solution, express $$x$$, $$y$$, and $$z$$ explicitly. If there are infinitely many solutions, express the variables in terms of the free variable $$z ($$since the problem specifies $$z$$ arbitrary for three variables).

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Key Concepts

Gauss-Jordan Elimination Method

Systems of Linear Equations and Solution Types

Parametric Representation of Solutions