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maw [93]
3 years ago
9

How do you know or what indicates to you that the solution set of an equation is all real numbers?

Mathematics
1 answer:
Harrizon [31]3 years ago
3 0

Answer:

  the equation is a tautology, or the system of equations is under-specified

Step-by-step explanation:

By definition, an equation that is a tautology is true for all values of the variable(s). Hence, its solution set is "all real numbers."

One such equation is ...

  x = x

__

An under-specified system of equations* may also have "all real numbers" as a solution set. For example, two equations in 3 unknowns:

  x + y = 3

  x + z = 5

The solution set is ...

  (x, y, z) = (x, 3-x, 5-x)

Every value of x will give a solution.

__

Similarly, a dependent system of equations may have "all real numbers" as a solution. A system is dependent when one or more of the equations can be derived from the others.

For example, adding the equation 2x+y+z=8 to the above set will give 3 equations in 3 unknowns. However, this added equation can be obtained by adding together the two above. It adds nothing that would restrict the solution set.

A system of two equations in two unknowns will be <em>dependent</em> if one of the equations is a constant multiple of the other. For example, x+y=1, 2x+2y=2 has a second equation that is 2 times the first. This set of equations has "all real numbers" as its solution set. (Of course, y = 1-x.)

_____

* Non-linear equations may impose limits on the variables, so that "all real numbers" cannot be a solution.

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