This problem addresses some common algebraic errors. For the equalities ststed below assume that x and y stand for real numbers. Assume that any denominatiors are non-zero. Mark the equalities with T(true) if they are true for all values of x and y, and F (false) otherwise. x/(x+y)=1/(1+y)

Let's analyze the given equality \( \frac{x}{x+y} = \frac{1}{1+y} \).

To determine whether this equation is true for all real numbers \( x \) and \( y \), we can test the equality by manipulating it algebraically.

### Step 1: Cross-Multiply
We start by cross-multiplying to eliminate the fractions:

\[
x(1 + y) = (x + y) \cdot 1
\]

This simplifies to:

\[
x + xy = x + y
\]

### Step 2: Simplify the Equation
Next, subtract \( x \) from both sides:

\[
xy = y
\]

### Step 3: Factor the Equation
Now, factor the right side:

\[
y(x - 1) = 0
\]

### Step 4: Analyze the Result
For this equation \( y(x - 1) = 0 \) to hold true for all real numbers \( x \) and \( y \), either \( y = 0 \) or \( x = 1 \). However, this is not true for all values of \( x \) and \( y \) because it only holds when \( y = 0 \) or \( x = 1 \).

Since the equation doesn't hold for all real numbers \( x \) and \( y \), the original equality is **false**.

### Conclusion
The given equality \( \frac{x}{x+y} = \frac{1}{1+y} \) is marked as **F** (false).

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Would you like more details on this, or do you have any other questions?

Here are 5 related questions to explore further:

1. What would happen if we assume \( y \) is constant and \( x \) varies?
2. Can we find a specific pair of values \( (x, y) \) that make the original equality true?
3. How does the inequality change if we add or subtract the same number on both sides?
4. What other types of algebraic errors are common in fraction manipulation?
5. How would the analysis change if we were working with complex numbers instead of real numbers?

**Tip:** When dealing with algebraic fractions, cross-multiplying is a powerful tool, but always check for extraneous solutions that might be introduced by this step.

Learn how to solve the algebraic equation x/(x+y) = 1/(1+y) with a detailed step-by-step solution. This problem covers key algebraic concepts, including fraction manipulation and cross-multiplication. Suitable for students in Grades 9-12.

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