To find an expression for [tex]\( y \)[/tex] in terms of [tex]\( b \)[/tex], we start with the given equations:
[tex]\[
y = \frac{2}{1 - x}
\][/tex]
[tex]\[
x = \frac{1}{b}
\][/tex]
First, substitute [tex]\( x \)[/tex] with [tex]\( \frac{1}{b} \)[/tex] in the expression for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{2}{1 - \frac{1}{b}}
\][/tex]
To simplify the denominator, let's combine terms inside the fraction:
[tex]\[
1 - \frac{1}{b}
\][/tex]
We need a common denominator to combine the terms. The common denominator is [tex]\( b \)[/tex], so we rewrite [tex]\( 1 \)[/tex] as [tex]\( \frac{b}{b} \)[/tex]:
[tex]\[
1 - \frac{1}{b} = \frac{b}{b} - \frac{1}{b} = \frac{b - 1}{b}
\][/tex]
Now, substitute this back into the expression for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{2}{\frac{b - 1}{b}}
\][/tex]
To divide by a fraction, we multiply by its reciprocal. The reciprocal of [tex]\( \frac{b - 1}{b} \)[/tex] is [tex]\( \frac{b}{b - 1} \)[/tex]:
[tex]\[
y = 2 \times \frac{b}{b - 1} = \frac{2b}{b - 1}
\][/tex]
Thus, the expression for [tex]\( y \)[/tex] in terms of [tex]\( b \)[/tex] is:
[tex]\[
y = \frac{2b}{b - 1}
\][/tex]
This is the required expression for [tex]\( y \)[/tex] in the simplest form.
