To determine which expression is equivalent to [tex]\(\frac{15}{x-6} + \frac{7}{z+6}\)[/tex], let's go through the following steps in detail:
1. Identify the common denominator:
The denominators in the given fractions are [tex]\(x - 6\)[/tex] and [tex]\(z + 6\)[/tex]. To combine these fractions, we need a common denominator. The common denominator for [tex]\(x - 6\)[/tex] and [tex]\(z + 6\)[/tex] is [tex]\((x - 6)(z + 6)\)[/tex].
2. Rewrite each fraction using the common denominator:
We must rewrite each fraction so that they both have this common denominator. To do this, we multiply the numerator and the denominator of each fraction by the respective missing term:
[tex]\[
\frac{15}{x-6} = \frac{15(z+6)}{(x-6)(z+6)}
\][/tex]
[tex]\[
\frac{7}{z+6} = \frac{7(x-6)}{(x-6)(z+6)}
\][/tex]
3. Combine the fractions:
Now that both fractions have the same denominator, we can add them together:
[tex]\[
\frac{15(z+6)}{(x-6)(z+6)} + \frac{7(x-6)}{(x-6)(z+6)}
\][/tex]
Combine the numerators:
[tex]\[
\frac{15(z+6) + 7(x-6)}{(x-6)(z+6)}
\][/tex]
4. Simplify the numerator:
Expand the numerator:
[tex]\[
15(z+6) + 7(x-6) = 15z + 90 + 7x - 42
\][/tex]
Combine like terms:
[tex]\[
15z + 90 + 7x - 42 = 7x + 15z + 48
\][/tex]
So, the expression becomes:
[tex]\[
\frac{7x + 15z + 48}{(x-6)(z+6)}
\][/tex]
5. Compare with given choices:
The simplified expression we obtained is:
[tex]\[
\frac{7x + 15z + 48}{(x-6)(z+6)}
\][/tex]
Among the choices provided, the one that matches most closely with our result is:
D. [tex]\(\frac{22x + 48}{x^2 - 36}\)[/tex]. Since none of the given options exactly match the derived numerator structure [tex]\(7x + 15z + 48\)[/tex], and considering the possibility that the denominator simplification might involve a typo or is expected to be interpreted adaptively to this very context, choice D is selected due to its similarity in structure when simplifying denominators.
