Verify if (x+7)/(7+x) = 1: Fraction Equality Check

Q: Why does x + 7 equal 7 + x?

The commutative property of addition tells us that changing the order of terms doesn't change their sum. Just like 3 + 5 = 5 + 3 = 8, we have x + 7 = 7 + x for any value of x.

Q: Does this work with subtraction too?

No! Subtraction is not commutative. For example, x - 7 ≠ 7 - x. So $ \frac{x-7}{7-x} $ does not equal 1. Only addition and multiplication are commutative.

Q: When does any fraction equal 1?

A fraction equals 1 when its numerator and denominator are identical (and non-zero). Examples: $ \frac{5}{5} = 1 $, $ \frac{2x+3}{2x+3} = 1 $ (when 2x + 3 ≠ 0).

Fraction Simplification with Commutative Property

Determine if the simplification described below is correct:

$\frac{x+7}{7+x}=1$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine if the reduction is correct

00:03 We'll use the substitution law and arrange the denominator to match the numerator

00:12 We'll reduce what we can, when reducing the entire fraction 1 always remains

00:18 Let's compare the expressions

00:23 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:

$\frac{x+7}{7+x}=1$

Step-by-step solution

To determine if the simplification $\frac{x+7}{7+x}=1$ is correct, we need to analyze the given expression.

The expression $\frac{x+7}{7+x}$ involves two terms: $x + 7$ in the numerator and $7 + x$ in the denominator. These are both algebraic expressions that involve addition.

In mathematics, the commutative property of addition tells us that the order of terms does not affect their sum. Therefore:

$x + 7$ is equivalent to $7 + x$ .

This means that the numerator and denominator are indeed the same expression. As a result, the fraction $\frac{x+7}{7+x}$ simplifies to $1$ , since any non-zero number divided by itself equals $1$ .

Hence, the simplification described is indeed correct.

The conclusion is that the simplification $\frac{x+7}{7+x}=1$ is Correct.

Final Answer

Correct

Key Points to Remember

Essential concepts to master this topic

Commutative Property: Addition order doesn't affect sum: x + 7 = 7 + x
Technique: Recognize identical numerator and denominator: $\frac{x+7}{7+x}$
Check: Any non-zero number divided by itself equals 1 ✓

Common Mistakes

Avoid these frequent errors

Not recognizing equivalent expressions
Don't treat x + 7 and 7 + x as different expressions = wrong conclusion that the fraction doesn't equal 1! This ignores the commutative property of addition. Always remember that addition order doesn't matter: a + b = b + a.

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 does x + 7 equal 7 + x?

The commutative property of addition tells us that changing the order of terms doesn't change their sum. Just like 3 + 5 = 5 + 3 = 8, we have x + 7 = 7 + x for any value of x.

What if x = -7? Wouldn't the fraction be undefined?

Great observation! When x = -7, both the numerator and denominator equal 0, making the fraction undefined. The simplification $\frac{x+7}{7+x} = 1$ is only valid when x ≠ -7.

How can I be sure the numerator and denominator are the same?

Look carefully at both expressions: x + 7 (numerator) and 7 + x (denominator). They contain the exact same terms, just in different order. Since addition is commutative, they're identical!

Does this work with subtraction too?

No! Subtraction is not commutative. For example, x - 7 ≠ 7 - x. So $\frac{x-7}{7-x}$ does not equal 1. Only addition and multiplication are commutative.

When does any fraction equal 1?

A fraction equals 1 when its numerator and denominator are identical (and non-zero). Examples: $\frac{5}{5} = 1$ , $\frac{2x+3}{2x+3} = 1$ (when 2x + 3 ≠ 0).

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Factorization questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime