Chapter 4 System of Linear Equations

Section 4.2 LS in two Variables

4.2.3 Addition Method

We will now discuss another, third method for solving systems of linear equations algebraically. But this method will develop its full potential only for larger systems, i.e. systems of many equations in many variables since it can be systematised very well. Here, we will discuss the general approach. First, let us see an example.
As for the other methods,the approach here is not uniquely defined: for example, equation $(1)$ could have been multiplied by $3$ and equation $(2)$ by $(-2)$

${\displaystyle \begin{array}{ccccccccc}\hfill (2x+y)·3& \hfill =\hfill & 9·3\hfill & \hfill \iff \hfill & \hfill 6x+3y& \hfill =\hfill & 27\hfill & \hfill \hfill & :\hspace{0.5em}\text{equation}\hspace{0.5em}(1\text{'}\text{'})\hspace{0.5em},\hfill \\ \hfill (3x-11y)·(-2)& \hfill =\hfill & 1·(-2)\hfill & \hfill \iff \hfill & \hfill -6x+22y& \hfill =\hfill & -2\hfill & \hfill \hfill & :\hspace{0.5em}\text{equation}\hspace{0.5em}(2\text{'}\text{'})\hspace{0.5em}.\hfill \end{array}}$

In the subsequent addition of equation $(1\text{'}\text{'})$ and equation $(2\text{'}\text{'})$ the variable $x$ could have been eliminated:

${\displaystyle 6x+3y-6x+22y=27-2\iff 25y=25\iff y=1\hspace{0.5em}.}$

To get the solution for $x$, the solution for $y$ then could have been substituted, e.g.  into equation $(2)$

${\displaystyle 3x-11·1=1\iff 3x=12\iff x=4\hspace{0.5em}.}$