Verify the Equation: Is (x+6)/(y+6) = x/y Correctly Simplified?

Q: Why can't I cancel the 6s from the numerator and denominator?

You can only cancel factors (numbers that multiply), not terms (numbers that add or subtract). The 6s are being added, so they're part of larger expressions that can't be simplified by canceling.

Q: How can I check if this equation is ever true?

Try specific numbers! If x = 2 and y = 3, then $ \frac{2+6}{3+6} = \frac{8}{9} $ but $ \frac{2}{3} = \frac{6}{9} $. Since $ \frac{8}{9} \neq \frac{6}{9} $, the equation is false.

Q: When can I actually cancel terms in fractions?

You can cancel when the same factor multiplies both the entire numerator and denominator. For example: $ \frac{6x}{6y} = \frac{x}{y} $ because 6 multiplies both x and y completely.

Q: How do I avoid this mistake in the future?

Remember: Only factors can be canceled, never terms! Before canceling anything, ask yourself: 'Is this number multiplying the entire numerator and denominator?' If not, don't cancel.

Fraction Simplification with Addition Terms

Determine if the simplification described below is correct:

$\frac{x+6}{y+6}=\frac{x}{y}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine if the reduction is correct

00:06 There's nothing to reduce, so let's compare the expressions as they are

00:14 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Determine if the simplification described below is correct:

$\frac{x+6}{y+6}=\frac{x}{y}$

Step-by-step solution

We use the formula:

$\frac{x+z}{y+z}=\frac{x+z}{y+z}$

$\frac{x+6}{y+6}=\frac{x+6}{y+6}$

Therefore, the simplification described is incorrect.

Final Answer

Incorrect

Key Points to Remember

Essential concepts to master this topic

Rule: Fractions with different terms in numerator and denominator don't cancel
Technique: Test with numbers: $\frac{2+6}{3+6} = \frac{8}{9} \neq \frac{2}{3}$
Check: Substitute specific values to verify if equation holds true ✓

Common Mistakes

Avoid these frequent errors

Incorrectly canceling terms that are added
Don't cancel the 6s from $\frac{x+6}{y+6}$ to get $\frac{x}{y}$ = wrong simplification! You can only cancel factors that multiply the entire numerator and denominator, not terms that are added. Always check if terms multiply the whole fraction before canceling.

Practice Quiz

Test your knowledge with interactive questions

Determine if the simplification shown below is correct:

$\frac{7}{7\cdot8}=8$

FAQ

Everything you need to know about this question

Why can't I cancel the 6s from the numerator and denominator?

You can only cancel factors (numbers that multiply), not terms (numbers that add or subtract). The 6s are being added, so they're part of larger expressions that can't be simplified by canceling.

How can I check if this equation is ever true?

Try specific numbers! If x = 2 and y = 3, then $\frac{2+6}{3+6} = \frac{8}{9}$ but $\frac{2}{3} = \frac{6}{9}$ . Since $\frac{8}{9} \neq \frac{6}{9}$ , the equation is false.

When can I actually cancel terms in fractions?

You can cancel when the same factor multiplies both the entire numerator and denominator. For example: $\frac{6x}{6y} = \frac{x}{y}$ because 6 multiplies both x and y completely.

Are there any values where this equation might be true?

Yes! While the equation isn't generally true, there might be specific values of x and y where both sides happen to equal the same number. But this doesn't make the simplification correct.

How do I avoid this mistake in the future?

Remember: Only factors can be canceled, never terms! Before canceling anything, ask yourself: 'Is this number multiplying the entire numerator and denominator?' If not, don't cancel.

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