To solve the equation [tex]\( d = \frac{5}{b} + \frac{9}{c} \)[/tex] for [tex]\( c \)[/tex], we will isolate [tex]\( c \)[/tex] through a series of algebraic manipulations. Here is a step-by-step explanation:
1. Subtract [tex]\(\frac{5}{b}\)[/tex] from both sides to isolate the term involving [tex]\( c \)[/tex]:
[tex]\[
d - \frac{5}{b} = \frac{9}{c}
\][/tex]
2. Take the reciprocal of both sides of the equation so as to solve for [tex]\( c \)[/tex]:
[tex]\[
\frac{1}{d - \frac{5}{b}} = \frac{c}{9}
\][/tex]
3. Multiply both sides by 9 to completely isolate [tex]\( c \)[/tex]:
[tex]\[
c = \frac{9}{d - \frac{5}{b}}
\][/tex]
4. Simplify the denominator:
To simplify the denominator [tex]\(d - \frac{5}{b}\)[/tex], we recognize that it can be rewritten in a single fractional term by finding a common denominator.
[tex]\[
d - \frac{5}{b} = \frac{db - 5}{b}
\][/tex]
5. Substitute this back into the expression for [tex]\( c \)[/tex]:
[tex]\[
c = \frac{9}{\frac{db - 5}{b}}
\][/tex]
6. Simplify the complex fraction by multiplying numerator and denominator by [tex]\( b \)[/tex]:
[tex]\[
c = 9 \cdot \frac{b}{db - 5} = \frac{9b}{db - 5}
\][/tex]
Therefore, the solution for [tex]\( c \)[/tex] is:
[tex]\[
c = \frac{9b}{db - 5}
\][/tex]
