Question

Solve each system, using the method indicated. 5x + 2y = -10 3x - 5y = -6 (Gauss-Jordan)

Accepted Answer

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

$$
\begin{cases}
5x + 2y = -10 \
3x - 5y = -6
\end{cases}
$$

The augmented matrix is:

$$
\left[\begin{array}{cc|c}
5 & 2 & -10 \
3 & -5 & -6
\end{array}ight]
$$ Use row operations to get a leading 1 in the first row, first column (pivot position). You can do this by dividing the entire first row by 5:

$$
R_1 	o \frac{1}{5} R_1
$$

Resulting in:

$$
\left[\begin{array}{cc|c}
1 & \frac{2}{5} & -2 \
3 & -5 & -6
\end{array}ight]
$$ Eliminate the entry below the pivot (make the element in the second row, first column zero) by replacing the second row with:

$$
R_2 	o R_2 - 3 	imes R_1
$$

This operation will help create a zero below the leading 1 in the first column. Next, create a leading 1 in the second row, second column (pivot position) by dividing the second row by the coefficient in that position. This will give you a 1 in the second row, second column. Finally, eliminate the entry above the second pivot (make the element in the first row, second column zero) by replacing the first row with:

$$
R_1 	o R_1 - (	ext{coefficient in } R_1, C_2) 	imes R_2
$$

This will put the matrix in reduced row echelon form, from which you can read off the values of $$x$$ and $$y.$$

Solve each system, using the method indicated.
5x + 2y = -10
3x - 5y = -6 (Gauss-Jordan)

Verified video answer for a similar problem:

Key Concepts

Systems of Linear Equations

Gauss-Jordan Elimination Method

Augmented Matrix Representation