Extraneous solution
Identify extraneous solutions in Grade 9 Algebra that satisfy a transformed equation but not the original. Always substitute back to verify each solution is valid.
Key Concepts
Property An extraneous solution is a solution that is acquired through the solving process, but makes a denominator in the original equation equal to 0. Explanation Watch out for these impostor solutions! They seem to solve the equation, but they are mathematical outlaws. Plugging them back into the original equation makes a denominator zero, which breaks the fundamental rules of fractions. Always check your answers to expose these fakes! Examples Solve $\frac{x+3}{x 4} = \frac{2x+1}{x 4}$. Cross multiplying gives $x+3=2x+1$, so $x=2$. This solution is valid. Solve $\frac{x 5}{x 5} = \frac{2x 9}{x 5}$. This simplifies to $1 = 2x 9$, so $x=5$. But $x=5$ makes the denominator 0, so it's an extraneous solution.
Common Questions
What is an extraneous solution in algebra?
An extraneous solution satisfies the equation after algebraic manipulation (such as squaring or multiplying by a variable expression) but does not satisfy the original equation. It must be identified and rejected.
How do you detect an extraneous solution?
After solving, substitute each answer into the original equation and check if both sides are equal. If the original equation is undefined or produces a false statement for that value, it is extraneous and must be discarded.
Which types of equations most commonly produce extraneous solutions?
Radical equations (where you square both sides), rational equations (where you multiply by a variable denominator), and logarithmic equations most commonly produce extraneous solutions. Always check answers in the original equation for these types.