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Use the Gauss-Jordan method to solve the system of equations. For systems in three variables with infinitely many solutions, give the solution with $z$ arbitrary; for any such equations with four variables, let $w$ be the arbitrary variable.

$\begin{aligned} &2 x-3 y=10 \\ &2 x+2 y=5 \end{aligned}$

Solution

VerifiedThe *Gauss-Jordan Method* is a method to solve a system of equations by using a matrix. It is done by doing elementary row operations to reduce the matrix into its diagonal form. The diagonal form of a matrix is where the only non-zero entry are the constants on the last column and $1$ where it is arranged diagonally in the matrix and any other entry is $0$. For instance, the diagonal form of a matrix with size $3\times 4$ is given as:

$\begin{bmatrix} 1 & 0 & 0 & p\\ 0 & 1 & 0 & q\\ 0 & 0 & 1 & r\\ \end{bmatrix}$

and its corresponding equations are:

$\begin{aligned} x&=p\\ y&=q\\ z&=r\\ \end{aligned}$

, where $r$ is the appointed variable for the first column, $y$ is the appointed variable for the second column, and $z$ is the appointed variable for the third column.

Note that the elementary matrix row operations are:

Operation $I$: Switch any rows in the matrix with another row.

Operation $II$: Multiply or divide any row in the matrix by any non-zero real number.

Operation $III$: Change the entries in a row by the sum of the current row that is to be replaced by another row in the matrix.

Hence, to solve a system of equations with a matrix, transform the matrix into its diagonal form by using the *Gauss-Jordan Method*.

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