Gauss-Jordan elimination is a technique that can be used to solve a system of linear equations. Here we use this technique to solve a system of two linear equations with two variables.

Unique solution

The system of linear equations may have a unique solution, for example:

• –x  + 2y = –3
• 4x + 2y = 2

1. Write the coefficients in an array format:
   [latex]\begin{array}{cc|c}-1&2&-3\\4&2&2\end{array}[/latex]
2. Obtain a 1 in the first row first column by dividing the first row by –1:
   [latex]\begin{array}{cc|c}\mathbf{1}&-2&3\\4&2&2\end{array}[/latex]
3. Obtain a 0 in the second row first column by subtracting four times the first row from the second row:
   [latex]\begin{array}{cc|c}1&-2&3\\\mathbf{0}&10&-10\end{array}[/latex]
4. Obtain a 1 in the second row second column by dividing the second row by 10:
   [latex]\begin{array}{cc|c}1&-2&3\\0&\mathbf{1}&-1\end{array}[/latex]
5. Obtain a 0 in the first row second column by adding two times the second row to the first row:
   [latex]\begin{array}{cc|c}1&\mathbf{0}&1\\0&1&-1\end{array}[/latex]

The solution to the system of linear equations is x = 1, y = –1, which is geometrically equivalent to two non-parallel straight lines meeting at a single point. We can check this is correct:

• –1  + 2(–1) = –1 – 2 = –3
• 4(1) + 2(–1) = 4 –2 = 2

No solution

The system of linear equations may have no solution, for example:

• –x  + 2y = –3
• 2x – 4y = 2

1. Write the coefficients in an array format:
   [latex]\begin{array}{cc|c}-1&2&-3\\2&-4&2\end{array}[/latex]
2. Obtain a 1 in the first row first column by dividing the first row by –1:
   [latex]\begin{array}{cc|c}\mathbf{1}&-2&3\\2&-4&2\end{array}[/latex]
3. Obtain a 0 in the second row first column by subtracting two times the first row from the second row:
   [latex]\begin{array}{cc|c}1&-2&3\\\mathbf{0}&0&-4\end{array}[/latex]
4. Stop because the second row is equivalent to 0x + 0y = –4, which has no solution.

This is geometrically equivalent to two parallel straight lines never meeting.

Infinite solutions

The system of linear equations may have infinite solutions, for example:

• –x  + 2y = –3
• 2x – 4y = 6

1. Write the coefficients in an array format:
   [latex]\begin{array}{cc|c}-1&2&-3\\2&-4&6\end{array}[/latex]
2. Obtain a 1 in the first row first column by dividing the first row by –1:
   [latex]\begin{array}{cc|c}\mathbf{1}&-2&3\\2&-4&6\end{array}[/latex]
3. Obtain a 0 in the second row first column by subtracting two times the first row from the second row:
   [latex]\begin{array}{cc|c}1&-2&3\\\mathbf{0}&0&0\end{array}[/latex]
4. Stop because the second row is equivalent to 0x + 0y = 0, which has an infinite number of solutions.

This is geometrically equivalent to two parallel straight lines that coincide. The solution set to the system of linear equations can be parameterized by setting y=t, so that x = 3 + 2t. We can check this is correct:

• –(3 + 2t) + 2t = –3 –2t + 2t = –3
• 2(3 + 2t) – 4t = 6 + 4t – 4t = 6