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Welcome to our Math lesson on Identify Conditional Identities and Inconsistent Equations, this is the third 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.
It is not appropriate to substitute values in a given mathematical sentence to check whether it is an identity, conditional or inconsistent equation because it may give a true result for some values and you may think it is an identity but there are other values that you have not considered which make the sentence false. Therefore, it is better to find a general form of all these three types of mathematical sentences; then check which category the given sentence does belong to.
From above, it is clear that the general form of identities (after having made all possible operations) is
where n is the order (highest power) of the original equation. In first - order identities, we have n = 1, in second - order ones n = 2 and so on.
In other words, all variables, coefficients and constants become zero after all operations and simplification are made. For example, in the equation
we have
Adding -3x and 2 to both sides to remove all terms from the right side yields
Hence, this is an identity.
On the other hand, if we have an equation that, after having made all possible operations and simplifications, takes the form
where a and b are numbers (a is a coefficient and b is a constant) and b is a variable, then we are dealing with a conditional identity, which is nothing more than a normal equation, like those we discussed in prior tutorials. Such equations have a limited maximum number of solutions depending on their order and number of variables. Thus, the first - order equations with one variable have at maximum one solution, the second - order equations with one variable (quadratic equations) have two solutions at maximum, and so on.
For example, 7x - 21 = 0 has a single solution (x = 3), because 7 · 3 - 21 = 0. All the other numbers give false results in regard to the equality between the two sides of the equation.
Last, if we have an equation that after having made all possible operations and simplifications gives a general form
where b is a constant, then we are dealing with an inconsistent equation, i.e. with an equation that has no solutions.
For example, 3x - 4 = - 2x + 1 + 5x is an inconsistent equation as after doing the operations, we obtain
(Here, b = 5)
Find whether the following mathematical sentences are identities, conditional identities (equations) or inconsistent equations.
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