Set equal to each other

In the set equal to each other method, the equations are converted to an unknown variable. The converted equation is then set equal with 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. $3x=y-30$
   2. $y-6x=54$

2. Convert both equations to one variable

   1. $3x=y-30$   $|+30$
   2. $y-6x=54$   $|+6x$
   
   
   1. $\color{red}{y}=\color{green}{3x+30}$
   2. $\color{red}{y}=\color{green}{54+6x}$

3. Set equal and solve equations

   $\color{red}{y}=\color{red}{y}$
   $\color{green}{3x+30}=\color{green}{54+6x}$   $|-30$
   $3x=24+6x$   $|-6x$
   $-3x=24$   $|:(-3)$
   $x=\color{blue}{-8}$

4. Insert $x=\color{blue}{-8}$ in I or II and solve

   I.$3\color{blue}{x}=y-30$
   
   $3\cdot\color{blue}{-8}=y-30$
   $-24=y-30$   $|+30$
   $y=6$

5. Solution set:

   $S=\{-8|6\}$