The general solution of a linear equation

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=2x-3 #
#\blue a=0# and #b\ne0#	none	#4x+5=4x+2#
#\blue a=0# and #\green b=0#	every #x#	#4x+5=4x+5#

ProofWe explain why. Remember that: the equation is #\blue a\cdot x+\green b=0#.

case	solutions	proof
#\blue a\ne0#	exactly one: #x=−\dfrac{\green b}{\blue a}#	Subtract #\green b# from both sides and next, divide both sides by #\blue a#.
#\blue a=0# and #\green b\ne0#	none	The equation becomes #\green b=0# which is not true, irrespective of the choice of #x#
#\blue a=0# and #\green b=0#	every #x#	The equation becomes #\green b=0# which is true for every choice of #x#

Parallel and intersecting lines

We saw that with the help of the reduction rules every linear equation can be written as #\blue a \cdot x +\green b=0#. This can be done by subtracting the terms on the right hand side from the left hand side.

We also see that two intersecting lines, who do not have the same slope, get #\blue a \ne 0#.

Two parallel lines have the same slope, hence get a #\blue a=0#.

Intersecting lines

\[\begin{array}{rcl}4x+5&=&2x-3 \\2x+8&=&0\end{array}\]

Parallel lines

\[\begin{array}{rcl}4x+5&=&4x+2 \\ 3&=&0 \end{array}\]

Equal lines

\[\begin{array}{rcl}4x+5&=&4x+5 \\ 0&=&0 \end{array}\]

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#.

Solve the equation #6-5\cdot x=-6-5\cdot x#.

If the equation has one solution, write #x=a# with the correct value of #a#. Simplify #a# as much as possible.
If the equation has no solution, write "none".
If each value of #x# is a solution, write "all".

#none#

The given equation in #x# can be reduced to an equation without #x#:
\[\begin{array}{rclcl}
6-5\cdot x&=&-6-5\cdot x\\&&\phantom{xxx}\blue{\text{the given equation}}\\ 6&=&-6\\&&\phantom{xxx}\blue{\text{left and right }5\cdot x\text{ added}}
\end{array}\]
The last equation is incorrect, which means no value of #x# lives up to the given equation.

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