Question

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

Answer 1

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

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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.

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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 dependent 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.)

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