12Addition method (1/2)

The third method for solving LGS is the addition method.

The goal of this method is to remove a variable by cleverly adding/subtracting the equations of the linear system.

Step 0: Tidying up the equations and selecting variables

First, for your equations, bring all the unknowns (with their coefficients) to the same side.

Look at the equations $\mathrm{I}$ and $\mathrm{II}$ with a sharp eye.

For example, is one of the variables in the first equation an integer multiple of itself in the second equation?

Then select that variable.

In the following example, in $\mathrm{I}$ the $x$ is on both sides and in $\mathrm{II}$ both variablesare not together on one side.

Clean up.

$\def\arraystretch{1.25} \begin{array}{lrcrl}\mathrm{I}& 2x-3y&=&-x-5& |+x\\\mathrm{II}& x&=&6y+2& |-6y\\\qquad\\\mathrm{I}& 3x-3y&=&-5\\\mathrm{II}& x-6y&=&2\end{array}$

Here both variables are suitable! In step 1, you see why.

Step 1: Multiplying the equations

In each case you have to multiply/divide the equations $\mathrm{I}$ and/or $\mathrm{II}$ such that the coefficients of the selected variables are equal (best according to the least common multiple).

Here you do not have to pay attention to the sign, but only to the coefficient.