Systems of Linear Equations: Simple Systems of Linear Equations
The notion of parameter
In a previous chapter, we saw that variables can play different roles, like constant and unknown. The role of parameter, which we introduce below, comes close to constant.
In Linear equations with one unknown we saw that the equation #ax+b =0# has solution #x=-\frac{b}{a}# if #a\ne0#. Here, #a# and #b# are real numbers. Often these numbers are specified. But we can write down the equation and the solution with unspecified #a# and #b#. In this case #a# and #b# are called parameters.
Generally speaking, parameters are variables that occur in mathematical expressions, but they are not the unknowns to be solved in the equation. They represent real numbers that are not yet determined.
The word parameter is usually pronounced with an accent on the second syllable: parámeter is most widely accepted, but some people put the emphasis on another syllable.
The linear relation between Celsius #C# and Fahrenheit #F# from the introduction to Linear equations with one unknown is an example in which #C# and #F# can both be seen as the unknown and the other one as a parameter.
It is important to note that we have marked #x# as the unknown. Alternatively, we could have considered #a# as the unknown, which would lead to the solution #a=-\frac{b}{x}#, at least if #x\ne0#.
This way we can also solve an equation with two unknowns.
The linear equation #ax+by+c=0#, with unknowns #x# and #y#, where #a#, #b#, and #c# are numbers or parameters, can be solved in the following two ways:
- By considering #y# temporarily as a parameter: if we solve the linear equation with unknown #x#, we get #x=-\frac{b}{a}y-\frac{c}{a}#. This can only be true if #a\ne0#. The solutions are all pairs #\rv{x,y}# of the form #\rv{-\frac{b}{a}y-\frac{c}{a},y}#. Here the role of #y# as parameter becomes clear: for every value of #y# there is exactly one solution.
- By considering #x# temporarily as a parameter: if we solve the linear equation with unknown #y#, we see that #y=-\frac{a}{b}x-\frac{c}{b}#. This can only be true if #b\ne0#; hence the solutions are all pairs #\rv{x,y}# that have the form #\rv{x,-\frac{a}{b}x-\frac{c}{b}}#. Here, the role of #a# as parameter becomes clear: for every value of #x# there is exactly one solution.
If #a\ne0# and #b\ne0#, both solutions are valid. The first solution one considers #x# as a function of #y#, the second considers #y# as a function of #x#.
The vertical line, corresponding to #b=0#, only occurs in the first case, whereas the horizontal line, corresponding to #a=0#, only occurs in the second case.
Thus far, we have in given all solutions if #a\ne0# or #b\ne0#. The outcome is a line in the plane. If #a=0# and #b=0#, then the equation is #c=0#. Hence, two cases remain:
- If #c=0#, then each pair #\rv{x,y}# is a solution.
- If #c\ne0#, then there are no solutions.
We proceed as in solving a linear equation with unknown #y# . We hence see #x# as parameter.
\[ \begin{array}{rclcl} 5\cdot x+8\cdot y&=&3&\phantom{xxxxx}&\color{blue}{\text{the original equation}}\\
8\cdot y &=& 3- 5 x&\phantom{xxxxx}&\color{blue}{\text{terms without }y\text{ to the right hand side}}\\
y &=& {{3-5\cdot x}\over{8}}&\phantom{xxxxx}&\color{blue}{\text{after division by the coefficient }8\text{ of }y}
\end {array}\] The result is #y= {{3-5\cdot x}\over{8}}= -{{5}\over{8}}x + {{3}\over{8}} # .
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