Equation solving

In mathematics, equation solving is the problem of finding what values (numbers, functions, sets...) fulfill a condition stated as an equality (an equation). Usually, this condition involves expressions with variables (or unknowns), which are to be substituted by values in order for the equality to hold. More precisely, an equation involves some free variables.

Related Topics:
Mathematics - Numbers - Functions - Sets - Equation - Variables - Free variable

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In one general case, we have a situation such as

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:f(x0,...,xn)=c, c constant

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which has a set of solutions S in the form

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:{(a0,...,an)∈Tn|f(a0,...,an)=c}

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with T the domain of the function. Note that the set of solutions can be empty (there are no solutions), singleton (there is exactly 1 solution), finite (there are only n number of solutions), or infinite (there are always solutions).

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For example, an expression such as

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:3x+2y=21z

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can be solved by first modifying the equation in some way as to preserve the equality, such as subtracting both sides by 21z to obtain

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:3x+2y-21z=0

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Now, it occurs that in solving this equation, that there is not just one solution to this equation, but a infinite set of solutions, which can be written

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:{(x, y, z)|3x+2y-21z=0}.

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One particular solution is x = 20/3, y = 11, z = 2. In fact, this particular set of solutions describe a plane in three dimensions, which passes through the point (20/3, 11, 2).

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