Linear formulas and equations: Linear equations and inequalities
The general solution of a linear equation
The general solution of a linear equationEach linear equation with unknown #x# can be reduced to the form #\blue a\cdot x+\green b=0#. We identify three cases.
case

solutions

example

#\blue a\ne0#

exactly one: #x=−\dfrac{\green b}{\blue a}#

#4x+5=2x3 #

#\blue a=0# and #b\ne0#

none

#4x+5=4x+2# 
#\blue a=0# and #\green b=0#

every #x#

#4x+5=4x+5# 
You do not have to remember these rules, since the solutions can be found easily by means of reduction.
When reducing the three case of the equation #\blue a \cdot x+\green b=0# can be recognized as follows.
case

solutions

recognizing when reducing

#\blue a\ne0#

exactly one: #x=−\dfrac{\green b}{\blue a}#

The reduction results in the equation#x=−\dfrac{\green b}{\blue a}#. 
#\blue a=0# and #\green b\ne0#

none

The reduction results in an expression without #x#, that is not true. For example #3=0#.

#\blue a=0# and #\green b=0#

every #x#

The reduction results in an expression without #x#, that is always true. For example #0=0#. 
#none#
The given equation in #x# can be reduced to an equation without #x#:
\[\begin{array}{rclcl}
65\cdot x&=&65\cdot x\\&&\phantom{xxx}\blue{\text{the given equation}}\\ 6&=&6\\&&\phantom{xxx}\blue{\text{left and right }5\cdot x\text{ added}}
\end{array}\]
The last equation is incorrect, which means no value of #x# lives up to the given equation.
The given equation in #x# can be reduced to an equation without #x#:
\[\begin{array}{rclcl}
65\cdot x&=&65\cdot x\\&&\phantom{xxx}\blue{\text{the given equation}}\\ 6&=&6\\&&\phantom{xxx}\blue{\text{left and right }5\cdot x\text{ added}}
\end{array}\]
The last equation is incorrect, which means no value of #x# lives up to the given equation.
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