Elimination of variables

Elimination of variables converts an equation to an unknown variable. The converted equation is then inserted into the other equation. One obtains an equation with only one variable, which can be solved by transforming the equation equivalently.

Example

1. Given are two linear equations:

   1. $3x+y=4$
   2. $y+36=5x$

2. One equation is converted to a variable

   1. $3x+y=4$
   2. $y+36=5x$   $|-36$
   
   
   1. $3x+\color{red}{y}=4$
   2. $\color{red}{y}=\color{green}{5x-36}$

3. Insert converted equation into the other one and solve

   II in I
   $3x+\color{green}{(5x-36)}=4$
   $3x+5x-36=4$
   $8x-36=4$   $|+36$
   $8x=40$   $|:8$
   $x=\color{blue}{5}$

4. Insert $x=\color{blue}{5}$ in I or II and solve

   I. $3\color{blue}{x}+y=4$
   
   $3\cdot\color{blue}{5}+y=4$
   $15+y=4$   $|-15$
   $y=-11$

5. Solution set:

   $S=\{5|-11\}$