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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 zz arbitrary; for any such equations with four variables, let ww be the arbitrary variable.

2x3y=102x+2y=5\begin{aligned} &2 x-3 y=10 \\ &2 x+2 y=5 \end{aligned}


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The 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 11 where it is arranged diagonally in the matrix and any other entry is 00. For instance, the diagonal form of a matrix with size 3×43\times 4 is given as:

[100p010q001r]\begin{bmatrix} 1 & 0 & 0 & p\\ 0 & 1 & 0 & q\\ 0 & 0 & 1 & r\\ \end{bmatrix}

and its corresponding equations are:

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

, where rr is the appointed variable for the first column, yy is the appointed variable for the second column, and zz is the appointed variable for the third column.

Note that the elementary matrix row operations are:

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

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

Operation IIIIII: 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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