To simplify the complex fraction [tex]\(\frac{1 - \frac{1}{y} - \frac{6}{y^2}}{1 - \frac{4}{y} + \frac{3}{y^2}}\)[/tex] using Method II, we will follow these detailed steps:
1. Combine terms to form a single fraction in the numerator and the denominator:
For the numerator [tex]\(1 - \frac{1}{y} - \frac{6}{y^2}\)[/tex]:
[tex]\[
1 - \frac{1}{y} - \frac{6}{y^2} = \frac{y^2}{y^2} - \frac{y}{y^2} - \frac{6}{y^2} = \frac{y^2 - y - 6}{y^2}
\][/tex]
For the denominator [tex]\(1 - \frac{4}{y} + \frac{3}{y^2}\)[/tex]:
[tex]\[
1 - \frac{4}{y} + \frac{3}{y^2} = \frac{y^2}{y^2} - \frac{4y}{y^2} + \frac{3}{y^2} = \frac{y^2 - 4y + 3}{y^2}
\][/tex]
2. Rewrite the complex fraction as a single fraction:
[tex]\[
\frac{\frac{y^2 - y - 6}{y^2}}{\frac{y^2 - 4y + 3}{y^2}}
\][/tex]
3. Simplify by multiplying by the reciprocal of the denominator:
[tex]\[
\frac{y^2 - y - 6}{y^2} \div \frac{y^2 - 4y + 3}{y^2} = \frac{y^2 - y - 6}{y^2} \cdot \frac{y^2}{y^2 - 4y + 3} = \frac{y^2 - y - 6}{y^2 - 4y + 3}
\][/tex]
4. Factor both the numerator and the denominator:
The numerator [tex]\(y^2 - y - 6\)[/tex] factors into [tex]\((y - 3)(y + 2)\)[/tex]:
[tex]\[
y^2 - y - 6 = (y - 3)(y + 2)
\][/tex]
The denominator [tex]\(y^2 - 4y + 3\)[/tex] factors into [tex]\((y - 1)(y - 3)\)[/tex]:
[tex]\[
y^2 - 4y + 3 = (y - 1)(y - 3)
\][/tex]
5. Simplify the factored form by canceling common factors:
[tex]\[
\frac{(y - 3)(y + 2)}{(y - 1)(y - 3)}
\][/tex]
The term [tex]\((y - 3)\)[/tex] appears in both the numerator and denominator and can be canceled out:
[tex]\[
\frac{(y - 3)(y + 2)}{(y - 1)(y - 3)} = \frac{y + 2}{y - 1}
\][/tex]
Thus, the simplified form of the given complex fraction is:
[tex]\[
\frac{y + 2}{y - 1}
\][/tex]
