Functions: Lines and linear functions
The general solution of a linear equation
In general the solutions of the equation #a\cdot x+b=0# with unknown #x# can be found as follows:
| case | solution |
|
#a\ne0#
|
exactly one: #x=−\dfrac{b}{a}#
|
|
#a=0# and #b\ne0#
|
no real solution |
|
#a=0# and #b=0#
|
true for all real values of #x#
|
We will show you why. (The equation is #ax+b=0#.)
| case | solution | explanation |
|
#a\ne0#
|
exactly one: #x=−\dfrac{b}{a}#
|
Subtract #b# from both the left and right hand side, next divide both sides by #a#. |
|
#a=0# and #b\ne0#
|
no real solution |
The equation becomes #b=0# which is not possible, independent on the value of #x#
|
|
#a=0# and #b=0#
|
true for all real values of #x#
|
The equation becomes #b=0# which is true for any value of #x#
|
You do not need to memorize these rules, because the solutions can easily be found by reduction. The three cases can also be recognized in terms of lines, as we will see later. We give an example of each case.
We will come across examples in which more general equations can be reduced to linear ones.
#x=8#
To see this, we reduce the equation as follows.
\[\begin{array}{rclcl}-4 x+48&=&16&\phantom{x}&\color{blue}{\text{the term }6 x\text{ moved to the left hand side}}\\ -4 x &=&-32&\phantom{x}&\color{blue}{\text{the term }48\text{ moved to the right hand side}} \\ x &=&8&\phantom{x}&\color{blue}{\text{dividing by }-4\text{}}\tiny.\end{array}\]
Hence, the only solution to the equation is #x=8#.
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