### Systems of linear equations and matrices: Systems of Linear Equations

### Elementary operations on systems of linear equations

We now discuss a general system of \(m\) linear equations with \(n\) unknowns \(x_1, \ldots, x_n\) in the following form \[\left\{\;\begin{array}{lllllllll} a_{11}x_1 \!\!\!\!&+&\!\!\!\! a_{12}x_2 \!\!\!\!&+&\!\!\!\! \cdots \!\!\!\!&+&\!\!\!\! a_{1n}x_n\!\!\!\!&=&\!\!\!\!b_1\\ a_{21}x_1 \!\!&+&\!\! a_{22}x_2 \!\!&+&\!\! \cdots \!\!&+&\!\! a_{2n}x_n\!\!\!\!&=&\!\!\!\!b_2\\ \vdots &&\vdots &&&& \vdots&&\!\!\!\!\vdots\\ a_{m1}x_1 \!\!\!\!&+&\!\!\!\! a_{m2}x_2 \!\!\!\!&+&\!\!\!\! \cdots \!\!\!\!&+&\!\!\!\! a_{mn}x_n \!\!\!\! &=&\!\!\!\!b_m\end{array}\right.\] Here, all \(a_{ij}\) and \(b_i\) with \(1\le i\le m\) and \(1\le j\le n\) are real or complex numbers. We discuss the sweep method for solving such a system. The strategy is to apply the following elementary operations to systems of linear equations, so as to obtain a simpler system step by step:

Elementary operations on systems of linear equations

In addition to expanding brackets, simplifying and regrouping subexpressions, we distinguish the following three **elementary operations on systems of linear equations**:

- Multiplication of an equation (that is to say, both sides of the equation) by a number distinct from zero.
- Addition of a multiple of one equation to one of the other equations.
- Interchange of two equations.

We speak of an **elementary reduction** if all steps in the reduction are elementary operations (we always allow for the expansion of brackets, simplification and regrouping of subexpressions in each equation).

If a system of linear equations is an elementary reduction of another system, then the two systems are *equivalent*.

The statement below explains why elementary reduction works.

The following method can be employed to obtain a parametric representation of the general solution of the linear system of equations described above:

- By selecting an equation, say #i_1#, in which #x_1# occurs (that is, the coefficient #a_{i_1,1}# is distinct from #0#), and next subtracting appropriate multiples of this equation from the other equations, we can ensure that #x_1# disappears from all of the other equations of the system.
- By selecting another comparison, say #i_2#, in which #x_2# occurs (so #a_{i_2,2}\ne0#), we can similarly achieve that #x_2# will no longer be found in any other equation. If #x_2# does not appear in an equation distinct from #i_1#, we go straight to #x_3#. We continue this way until all unknown have been treated.
- The general solution of the system of equations in parameter form will appear in parameter form as follows: The equations #i_1# , #i_2,\ldots# (if any) are used to express #x_1#, #x_2,\ldots# (if any) in the other unknowns. These other unknowns can be chosen freely. In the general solution, they (or variables not used before) can act as free parameters.

Here are a few examples of systems with two or three unknowns.

- If there is no solution, write #none#
- If there is one solution, write #x=a\land y=b# for appropriate numbers #a# and #b#. You can find the #\land# on the mathematical input editor on the "function" tab.
- If there are multiple solutions, express one of the variables in terms of the other
- If all values of #x# and #y# are solutions, write #all#

To see this, we begin with the original system of equations \[\lineqs{ x-y&=&1\cr -2 x+3 y&=&-4 \cr}\] and we replace the second equation by the difference of the current and the multiple of the first equation that makes \(x\) disappear (add #2# times the first equation to the second). This way the system becomes

\[\lineqs{ x-y&=&1\cr y&=&-2 \cr}\] It follows from the second equation that \(y=-2\). Substitution of this value for \(y\) in the first equation turns that equation into a linear equation for \(x\) only, with solution \(x=-1\).

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