Basic algebra skills: Linear functions
The general solution of a linear equation
We have seen that we can solve linear equations by reduction. A linear equation has in general three solutions.
In general the solutions of the equation #a\cdot x+b=0# with unknown #x#, and #a# and #b# real numbers, 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#
|
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.
#x=8#
To see this, we reduce the equation as follows.
\[\begin{array}{rclcl}-3 x+32&=&8&\phantom{x}&\color{blue}{\text{the term }4 x\text{ moved to the left hand side}}\\ -3 x &=&-24&\phantom{x}&\color{blue}{\text{the term }32\text{ moved to the right hand side}} \\ x &=&8&\phantom{x}&\color{blue}{\text{dividing by }-3\text{}}\tiny.\end{array}\]
Hence, the only solution to the equation is #x=8#.
Or visit omptest.org if jou are taking an OMPT exam.
