Simplify (1-(2x)/y)/(1/x+x/y)

Solution:

Step 1:

Multiply both the top and bottom of the fraction by \(yx\).

Step 1.1:

Multiply \(\frac{1 - \frac{2x}{y}}{\frac{1}{x} + \frac{x}{y}}\) by \(\frac{yx}{yx}\) to get \(\frac{yx}{yx} \cdot \frac{1 - \frac{2x}{y}}{\frac{1}{x} + \frac{x}{y}}\).

Step 1.2:

Combine the terms to obtain \(\frac{yx(1 - \frac{2x}{y})}{yx(\frac{1}{x} + \frac{x}{y})}\).

Step 2:

Distribute \(yx\) across the terms in the numerator and denominator to get \(\frac{yx \cdot 1 - yx \cdot \frac{2x}{y}}{yx \cdot \frac{1}{x} + yx \cdot \frac{x}{y}}\).

Step 3:

Simplify by canceling out common factors.

Step 3.1:

Eliminate the common \(y\) factor.

Step 3.1.1:

Rewrite the negative term in the numerator to \(\frac{yx \cdot 1 - yx \cdot \frac{-2x}{y}}{yx \cdot \frac{1}{x} + yx \cdot \frac{x}{y}}\).

Step 3.1.2:

Factor out \(y\) from \(yx\) in the numerator.

Step 3.1.3:

Cancel out the \(y\) to get \(\frac{yx \cdot 1 - x(-2x)}{yx \cdot \frac{1}{x} + yx \cdot \frac{x}{y}}\).

Step 3.1.4:

Rewrite the expression as \(\frac{yx - 2x^2}{yx \cdot \frac{1}{x} + yx \cdot \frac{x}{y}}\).

Step 3.2:

Express \(x\) as \(x^1\).

Step 3.3:

Keep \(x\) as \(x^1\) in the term \(2x^2\).

Step 3.4:

Apply the exponent rule \(a^m a^n = a^{m+n}\) to combine like terms.

Step 3.5:

Add the exponents to get \(\frac{yx - 2x^{2}}{yx \cdot \frac{1}{x} + yx \cdot \frac{x}{y}}\).

Step 3.6:

Cancel the common \(x\) factor.

Step 3.6.1:

Factor \(x\) from \(yx\) in the denominator.

Step 3.6.2:

Cancel the \(x\) to get \(\frac{yx - 2x^2}{y + yx \cdot \frac{x}{y}}\).

Step 3.6.3:

Rewrite the expression as \(\frac{yx - 2x^2}{y + x^2}\).

Step 3.7:

Eliminate the common \(y\) factor in the denominator.

Step 3.7.1:

Factor \(y\) from \(yx\) in the denominator.

Step 3.7.2:

Cancel the \(y\) to get \(\frac{yx - 2x^2}{y + x^2}\).

Step 3.7.3:

Rewrite the expression as \(\frac{yx - 2x^2}{y + x^2}\).

Step 3.8:

Express \(x\) as \(x^1\) in the term \(x^2\).

Step 3.9:

Keep \(x\) as \(x^1\) in the term \(x^2\).

Step 3.10:

Apply the exponent rule \(a^m a^n = a^{m+n}\) to combine like terms.

Step 3.11:

Add the exponents to get \(\frac{yx - 2x^2}{y + x^2}\).

Step 4:

Simplify the numerator.

Step 4.1:

Factor \(x\) out of the numerator.

Step 4.1.1:

Factor \(x\) out of \(yx - 2x^2\) to get \(\frac{x(y - 2x)}{y + x^2}\).

Step 4.1.2:

Continue factoring \(x\) to obtain \(\frac{x(y - 2x)}{y + x^2}\).

Step 4.1.3:

Factor \(x\) completely from the numerator to get \(\frac{x(y - 2x)}{y + x^2}\).

Step 4.2:

Multiply \(y\) by \(1\) to finalize the expression as \(\frac{x(y - 2x)}{y + x^2}\).

Knowledge Notes:

The problem involves simplifying a complex fraction, which is a fraction where the numerator, the denominator, or both are also fractions. The steps to simplify such a fraction typically involve:

1. Multiplying the numerator and the denominator by a common term to eliminate the smaller fractions within. This is often the least common multiple (LCM) of the denominators of the smaller fractions.

2. Applying the distributive property to simplify the expression, which involves multiplying a term across a sum or difference within parentheses.

3. Canceling common factors, which means reducing terms that appear in both the numerator and the denominator.

4. Applying exponent rules, such as \(a^m \cdot a^n = a^{m+n}\), to simplify terms with exponents.

5. Factoring, which involves expressing a term as the product of its factors, which can then be canceled if they appear in both the numerator and the denominator.

In this problem, the process includes multiplying by \(yx\) to clear the fractions within the complex fraction, distributing \(yx\) across the terms, canceling common factors of \(y\) and \(x\), applying exponent rules, and factoring out common terms to simplify the expression to its simplest form.