Solving Systems with Gaussian Elimination: Learn It 1

  • Convert between augmented matrices and systems of equations
  • Perform row operations on a matrix.
  • Solve a system of linear equations using row operations.

Augmented Matrix of a System of Equations

A matrix can serve as a device for representing and solving a system of equations. To express a system in matrix form, we extract the coefficients of the variables and the constants, and these become the entries of the matrix. We use a vertical line to separate the coefficient entries from the constants, essentially replacing the equal signs. When a system is written in this form, we call it an augmented matrix.

A diagram showing a system of two linear equations being rewritten as an augmented matrix. The coefficients of x form the first column, the coefficients of y form the second column, and the constants form the third column.

  • Each column then would be the coefficients of one of the variables in the system or the constants.
  • A vertical line replaces the equal signs.

We call the resulting matrix the augmented matrix for the system of equations.

For example, consider the following [latex]2\times 2[/latex] system of equations.

[latex]\begin{array}{l}3x+4y=7\\ 4x - 2y=5\end{array}[/latex]

We can write this system as an augmented matrix:

[latex]\left[\begin{array}{cc|c}\hfill 3& \hfill 4& \hfill 7\\ \hfill 4& \hfill -2& \hfill 5\\ \end{array}\right][/latex]

We can also write a matrix containing just the coefficients. This is called the coefficient matrix.

[latex]\left[\begin{array}{cc}3& 4\\ 4& -2\end{array}\right][/latex]

A three-by-three system of equations such as

[latex]\begin{array}{l}3x-y-z=0\hfill \\ \text{ }x+y=5\hfill \\ \text{ }2x - 3z=2\hfill \end{array}[/latex]

has a coefficient matrix

[latex]\left[\begin{array}{rrr}\hfill 3& \hfill -1& \hfill -1\\ \hfill 1& \hfill 1& \hfill 0\\ \hfill 2& \hfill 0& \hfill -3\end{array}\right][/latex]

and is represented by the augmented matrix

[latex]\left[\begin{array}{ccc|c}\hfill 3& \hfill -1& \hfill -1& \hfill 0\\ \hfill 1& \hfill 1& \hfill 0& \hfill 5\\ \hfill 2& \hfill 0& \hfill -3& \hfill 2\\ \end{array}\right][/latex]

Notice that the matrix is written so that the variables line up in their own columns: [latex]x-[/latex]terms go in the first column, [latex]y-[/latex]terms in the second column, and [latex]z-[/latex]terms in the third column. It is very important that each equation is written in standard form [latex]ax+by+cz=d[/latex] so that the variables line up. When there is a missing variable term in an equation, the coefficient is [latex]0[/latex].

How To: Given a system of equations, write an augmented matrix

  1. Write the coefficients of the [latex]x-[/latex]terms as the numbers down the first column.
  2. Write the coefficients of the [latex]y-[/latex]terms as the numbers down the second column.
  3. If there are [latex]z-[/latex]terms, write the coefficients as the numbers down the third column.
  4. Draw a vertical line and write the constants to the right of the line.
Write the augmented matrix for the given system of equations.

[latex]\begin{array}{l}\text{ }x+2y-z=3\hfill \\ \text{ }2x-y+2z=6\hfill \\ \text{ }x - 3y+3z=4\hfill \end{array}[/latex]

We can use augmented matrices to help us solve systems of equations because they simplify operations when the systems are not encumbered by the variables. However, it is important to understand how to move back and forth between formats in order to make finding solutions smoother and more intuitive. Here, we will use the information in an augmented matrix to write the system of equations in standard form.

Find the system of equations from the augmented matrix.

[latex]\left[\begin{array}{ccc|c}\hfill 1& \hfill -3& \hfill -5& \hfill -2\\ \hfill 2& \hfill -5& \hfill -4& \hfill 5\\ \hfill -3& \hfill 5& \hfill 4& \hfill 6\\ \end{array}\right][/latex]