Simultaneous Equations With Non Unique Solutions
The set of simultaneous equations
has the solutions
and
and
Infact many values of
and
satisfyboth equations.
Notice that if the second equation is divided by 2 throughout, weobtain the first equation, so that these two equations are in factjust one independednt equation. In general, we need two independentlinear equations to solve simultaneous equations with two unknowns tofind unique solutions. If we only have one independent linearequation with two unknowns, then we will be able to find an infinitenumber of solutions.
In general, if we have n equations with n unknowns, we can onlyfind unique solutions if the equations are independent, so that noneof the equations can be expressed in terms of the others.
Consider the equations
The third equation is the sum of the first two, so this system ofequations is not independent and does not have a unique solution.
In fact, we can ignore one of the equations, and still solve thesystem. If we ignore the third equation, we have the system
Set
togive
(1)
(2)
(1)-(2) gives
(3)
Substitute this expression into (1) to give
(4)
Then the set of solutions is given by
These are parametric coordinates for a line. We can make thesolution more obvious.
Eliminate t between (3) and (4) by finding(3)*6+4:
isobvuously the equation of a line.
If we have three equations in three unknowns and each is amultiple of the other, then we can discard two and the result will beone of the equations which must be the equation of a plane.
Example:
The second is twice the first, and the third is three times thefirst, so two are redundant.
If we discard the last two, the solution set is the set ofsolutions toandthis is a plane.
