Addition method

The addition method adds or subtracts the equations, giving only one equation with one variable. To achieve this, one (or both) of the equations must be multiplied or divided by a value so that the coefficients of a variable in the equations are counterparts.

Example

1. Given are two linear equations:

   
   
   1. $2x-4y=-10$
   2. $5x+5y=20$

2. Multiply so that the coefficients of a variable are counterparts

   1. $2x-4y=-10$   $|\color{red}{\cdot5}$
   2. $5x+5y=20$   $|\color{red}{\cdot(-2)}$
   
   
   1. $\color{red}{10x}-20y=-50$
   2. $\color{red}{-10x}-10y=-40$

3. Addition of the two equations such that $x$ disappears

       I. $\color{red}{10x}-20y=-50$
   + II. $\color{red}{-10x}-10y=-40$
   
   III. $-30y=-90$

4. Solve equation III

   III. $-30y=-90$   $|:(-30)$
   $y=\color{blue}{3}$

5. Insert $y=\color{blue}{3}$ in I. or II. and solve

   $2x-4\color{blue}{y}=-10$
   
   $2x-4\cdot\color{blue}{3}=-10$
   $2x-12=-10$   $|+12$
   $2x=2$   $|:2$
   $x=1$

6. Solution set:

   $S=\{1|3\}$