extraneous solution

(noun)

An answer to an equation that emerges from the process of solving the problem but that is not a valid answer to the original problem.

Related Terms

  • extraneous solutions
  • square
  • root
  • radical

Examples of extraneous solution in the following topics:

  • Radical Equations

    • However, squaring both sides can introduce extraneous solutions (i.e., false answers), so it is important to check the answers after solving.
    • If no answer checks out, then the solution is "no solution."
    • Therefore, $x=17$ is a valid solution to the equation $\sqrt{6x-2}-3=7$.
    • This means that $10.2$ is an extraneous solution.
    • Because it is the only answer we found, the answer to this problem is "no solution."
  • Introduction to Systems of Equations

    • Some linear systems may not have a solution, while others may have an infinite number of solutions.
    • Even so, this does not guarantee a unique solution.
    • A solution to the system above is given by
    • An inconsistent system has no solution.
    • A dependent system has infinitely many solutions.
  • Equations in Two Variables

    • Equations in two variables have not one solution but a series of solutions that will satisfy the equation for both variables.
    • Each solution is an ordered pair and can be written in the form $(x, y)$.
    • For example, $(1, 2)$ is a solution to the equation.
    • Another solution is $(30, 60)$, because $(60) = 2(30)$.
    • Therefore, the solution is $(3, 5)$.
  • Graphing Inequalities

    • In our study of linear equations in two variables, we observed that all the solutions to an equation—and only those solutions— were located on the graph of that equation.
    • The resulting ordered pair will be one solution to the equation.
    • So, let's substitute $x = 0 $ to find one solution:
    • Graph showing all possible solutions of the given inequality.
    • The solutions lie in the shaded region, including the boundary line.
  • Inconsistent and Dependent Systems in Three Variables

    • Independent systems have a single solution.
    • Dependent systems have an infinite number of solutions.
    • Inconsistent systems have no solution.
    • An infinite number of solutions can result from several situations.
    • The three planes could be the same, so that a solution to one equation will be the solution to the other two equations.
  • Inconsistent and Dependent Systems

    • Two properties of a linear system are consistency (are there solutions?
    • A solution to a linear system is an assignment of numbers to the variables such that all the equations are simultaneously satisfied.
    • A solution to the system above is given by
    • A linear system is consistent if it has a solution, and inconsistent otherwise.
    • Note that any two of these equations have a common solution.
  • Inconsistent and Dependent Systems in Two Variables

    • For linear equations in two variables, inconsistent systems have no solution, while dependent systems have infinitely many solutions.
    • An independent system of equations has exactly one solution $(x,y)$.
    • An inconsistent system has no solution, and a dependent system has an infinite number of solutions.
    • Note that there are an infinite number of solutions to a dependent system, and these solutions fall on the shared line.
    • A linear system is consistent if it has a solution, and inconsistent otherwise.
  • Graphs of Linear Inequalities

    • To find solutions for the group of inequalities, observe where the area of all of the inequalities overlap.
    • These overlaps of the shaded regions indicate all solutions (ordered pairs) to the system.
    • This also means that if there are inequalities that don't overlap, then there is no solution to the system.
    • There are no solutions above the line.
    • The origin is a solution to the system, but the point $(3,0)$ is not.
  • What is an Equation?

    • In an equation with one variable, the variable has a solution, or value, that makes the equation true.
    • The values of the variables that make an equation true are called the solutions of the equation.
    • Thus, we can easily check whether a number is a genuine solution to a given equation.
    • For example, let's examine whether $x=3$ is a solution to the equation  $2x + 31 = 37$.
    • Therefore, we can conclude that $x = 3$ is, in fact, a solution to the equation $2x+31=37$.
  • Solving Problems with Inequalities

    • Note that the expression x > 12 has infinitely many solutions.
    • Some solutions are: 13, 15, 90, 12.1, 16.3, and 102.51.
    • A linear equation, we know, may have exactly one solution, infinitely many solutions, or no solution.
    • Speculate on the number of solutions of a linear inequality.
    • A linear inequality may have infinitely many solutions or no solutions.
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