Elementary Algebra - Solving Algebraic Equations - Relation Between Solvability and Multiplicity

Relation Between Solvability and Multiplicity

Given any system of linear equations, there is a relation between multiplicity and solvability.

If one equation is a multiple of the other (or, more generally, a sum of multiples of the other equations), then the system of linear equations is undetermined, meaning that the system has infinitely many solutions. Example:

begin{cases} begin{align} x + y &= 2 \
2x + 2y &= 4 end{align}end{cases}

has solutions for such as (1, 1), (0, 2), (1.8, 0.2), (4, −2), (−3000.75, 3002.75), and so on.

When the multiplicity is only partial (meaning that for example, only the left hand sides of the equations are multiples, while the right hand sides are not or not by the same number) then the system is unsolvable. For example, in

begin{cases}begin{align}x + y & = 2 \
4x + 4y &= 1 end{align}end{cases}

the second equation yields that which is in contradiction with the first equation. Such a system is also called inconsistent in the language of linear algebra. When trying to solve a system of linear equations it is generally a good idea to check if one equation is a multiple of the other. If this is precisely so, the solution cannot be uniquely determined. If this is only partially so, the solution does not exist.

This, however, does not mean that the equations must be multiples of each other to have a solution, as shown in the sections above; in other words: multiplicity in a system of linear equations is not a necessary condition for solvability.

Read more about this topic:  Elementary Algebra, Solving Algebraic Equations

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