Solution

Pronunciation: /soʊˈlu ʃən/ ?

The solution to a math problem is the correct answer for the problem. The solution to an equation is a set of one or more values that, when substituted for a variable, make an equation or system true or consistent. To solve a math problem is to find the solution of the problem.

For example, given the equation x + 3 = 5, the solution is x = 2. Try substituting 2 for x in the equation. The equation becomes 2 + 3 = 5. Since this is a true statement, we know that 2 is a solution to the equation.

An extraneous solution is a apparent solution that does not check out. Extraneous solutions can occur when multiplying both sides of an equation by a variable, and by squaring both sides of an equation.

StepEquationDescription
1x-2=square root(x).This is the equation to solve.
2(x-2)^2=(square root(x))^2.Square both sides of the equation. This is the step where and extraneous solution is introduced. Expand both sides of the equation by applying the exponents.
3x-4x+4=x implies x-5x+4=0.Subtract x from both sides of the equation.
4x-5x+4=0 implies (x-4)(x-1)=0.Factor the left hand side of the equation.
5(x-4)(x-1)=0 implies x=4 or x=1.The factors imply solutions of 4 and 1.
6x-2=square root(x), x=4 implies 4-2=square root(2) implies 2=2.Check the solution x=4 by substituting 4 for x in the original equation. The equation now simplifies to 2=2. Since this is a true statement, 2 is a solution to the equation. 2 is not an extraneous solution to the equation.
7x-2=square root(x), x=1 implies 1-2=square root(1) implies -1=1.Check the solution x=1 by substituting 1 for x in the original equation. The equation now simplifies to -1=1. Since this is a false statement, 1 is a not solution to the equation. 1 is an extraneous solution.
Table 1: Finding an extraneous solution.

References

  1. solution. http://wordnet.princeton.edu/. WordNet. Princeton University. (Accessed: 2011-01-08). http://wordnetweb.princeton.edu/perl/webwn?s=solution&sub=Search+WordNet&o2=&o0=1&o7=&o5=&o1=1&o6=&o4=&o3=&h=.
  2. extraneous. http://wordnet.princeton.edu/. WordNet. Princeton University. (Accessed: 2011-01-08). http://wordnetweb.princeton.edu/perl/webwn?s=solution&sub=Search+WordNet&o2=&o0=1&o7=&o5=&o1=1&o6=&o4=&o3=&h=.

Cite this article as:


Solution. 2009-01-09. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/s/solution.html.

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Revision History


2009-01-09: Added section on 'extraneous solution' (McAdams, David.)
2008-12-03: Added 'one or more values' to the definition (McAdams, David.)
2008-09-15: Added 'More Information' (McAdams, David.)
2008-03-05: Add solution to a math problem (McAdams, David.)
2007-07-11: Initial version (McAdams, David.)

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