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Welcome to our Math lesson on Identities, Conditional Identities (Equations) and Inconsistent Equations, this is the second lesson of our suite of math lessons covering the topic of Identities, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
All equations we have considered so far had a fixed number of solutions (roots). In most cases, they had only one solution, i.e. they were true for only one value, while for the rest of the numbers the equations were false. For example, the equation
is true only for x = 5, as 2 · 5 - 1 = 10 - 1 = 9. If we try to insert another value to replace the variable x, we obtain a false equation, as the left part is not equal to the right one. For example, taking x = 3 yields 2 · 3 - 1 = 6 - 1 = 5, not 9 as it must be.
However, there are some special equations known as 'identities', which give a true result for whatever value of their variable(s). For example,
is an identity as whatever value we take for x, the left part will be always equal to the right part. Let's try some numbers for x. Thus, for x = 1, we have
For x = 3, we have
For x = 12, we have
For x = 0, we have
and so on.
On the other hand, equations that have a fixed number of roots are known as 'conditional identities', as they are only true for specific values of the variable(s). All equations discussed so far in previous tutorials, including the quadratic ones, belong to this type. For example, the equation
is a conditional identity, as it is true only for x = 3 but not for other values of the variable.
There is also a third category of equations known as 'inconsistent equations', which are always false for all values of their variable(s). For example, the equation
belongs to this type, as whatever value of x we take, gives not - true equality. For example, for x = 4, we have
and it is clear that this is not true. For x = 0, we have
which is not true either, and so on.
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