Extraneous Solutions
Extraneous Solutions addresses the phenomenon where multiplying an equation by a variable expression can introduce false solutions that satisfy the transformed equation but not the original. This concept is taught in Yoshiwara Elementary Algebra Chapter 8: Algebraic Fractions and is critical for Grade 6 students solving rational equations. Always substituting answers back into the original equation is the essential verification step to identify and discard extraneous solutions.
Key Concepts
Property When multiplying an equation by an expression containing a variable, a false solution may be introduced. Such a solution is called an extraneous solution. To check for extraneous solutions, substitute the possible solution into the original equation. If it causes any denominator in the original equation to equal zero, that solution is extraneous and must be discarded.
Examples Solve $5 + \frac{2}{x 4} = \frac{x 2}{x 4}$. Multiplying by the LCD $x 4$ gives $$5(x 4) + 2 = x 2$$, so $$5x 20 + 2 = x 2$$, which simplifies to $$4x = 16$$ and $$x=4$$. Since $x=4$ makes the original denominator zero, it is an extraneous solution. There is no solution. Solve $\frac{y}{y+2} 1 = \frac{ 2}{y+2}$. The LCD is $y+2$. This gives $$y 1(y+2) = 2$$, so $$y y 2 = 2$$, which is $$ 2= 2$$. The proposed solution is $y= 2$, which is extraneous. No solution. Solve $\frac{3a}{a 6} 4 = \frac{18}{a 6}$. The LCD is $a 6$. This gives $$3a 4(a 6) = 18$$, so $$3a 4a + 24 = 18$$, which simplifies to $$ a = 6$$ and $$a=6$$. This solution is extraneous.
Explanation An extraneous solution is a tricky fake out! It appears to be a valid answer, but it's invalid in the original context because it makes you divide by zero. Always plug your solution back in to check the original denominators.
Common Questions
What is an extraneous solution?
An extraneous solution is a value that satisfies a transformed version of the equation but makes the original equation undefined or false when substituted back.
Why do extraneous solutions appear?
They are introduced when both sides are multiplied by a variable expression (as in clearing denominators in rational equations), which can make the equation true for values that were originally excluded.
How do you identify an extraneous solution?
Substitute each potential solution back into the original equation. If it makes a denominator equal to zero or the equation false, it is extraneous and must be discarded.
Where are extraneous solutions covered in Yoshiwara Elementary Algebra?
This concept is in Chapter 8: Algebraic Fractions of Yoshiwara Elementary Algebra.
Do extraneous solutions appear in radical equations too?
Yes. Squaring both sides of a radical equation can also introduce extraneous solutions. Always check answers in the original equation.