Machtsfuncties en vergelijkingen *: De grafiek van y = 1/x *
Fractional linear equations
A fractional linear function is an expression of the form #\dfrac{ax+b}{cx+d}#, where #a#, #b#, #c#, and #d# are fixed numbers and #x# is a variable.
The equation \[\dfrac{ax+b}{cx+d} = w\] with unknown #x# is called a fractional linear equation.
The equation \[\dfrac{ax+b}{cx+d} = w\] with unkown #x# deals with the question for which #x# the function #\frac{ax+b}{cx+d}# with variable #x# attains the value #w#. In the chapter Functions the notion function will be discussed.
The fractional linear function is a quotient of the linear function #ax+b# by the linear function #cx+d#. The function is not defined if #x# satisfies #cx+d=0# (for, then the denominator equals #0#). So you need to solve a linear equation to find out where the function is defined.
The fractional linear equation \[\dfrac{ax+b}{cx+d} = w\] is equivalent to the system \[ax+b=w\cdot (cx+d)\land cx+d\ne0\tiny.\]
This can be seen by observing that the fraction \(c x + d=0\) is not defined if #cx+d=0# and by multiplying both sides by # c x +d#.
In this way the solving of a fractional linear equation is reduced to solving a linear equation and selecting the solution(s) #x# with #cx+d\ne0#.
It may happen that the fractional linear equation has no solution, but also that every #x# is a solution. Below you see the graphs of #y = \dfrac{ax+b}{cx+d}# and #y = w#. Solutions to #\dfrac{ax+b}{cx+d}= w# are the #x#-coordinates of point where the two graphs meet. Move the sliders to get an idea how many solutions of the equation #\dfrac{ax+b}{cx+d} = w# with unknown #x# to expect.
This is done by
- first multiplying the left and right hand side by #(-4 x -40)# ;
- #\;{-8 x -16} = -4 x -40#.
- then move all terms containing #x# to the left hand side and all constant terms to the right hand side:
- #(-8+4) x = -40+16#;
- next divide by #(-8+4)=-4# ;
- #\displaystyle x=6#
- and finally check that the solution #x=6# will not be zero in #-4 x -40# (the denominator in the left hand side of the equation #\;\dfrac{-8 x -16}{-4 x -40} = 1# cannot be equal to zero).
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