To solve the equation [tex]\(-8 - \frac{6}{7} = \frac{2}{3}\)[/tex] for [tex]\(a\)[/tex], follow these steps:
1. Combine the constants on the left-hand side:
[tex]\[
-8 - \frac{6}{7}
\][/tex]
To combine these, we need a common denominator for the fractions. The number 8 can be written as a fraction with 7 in the denominator:
[tex]\[
-8 = -\frac{56}{7}
\][/tex]
Therefore, the left-hand side of the equation now becomes:
[tex]\[
-\frac{56}{7} - \frac{6}{7} = -\frac{56 + 6}{7} = -\frac{62}{7}
\][/tex]
2. Rewrite the equation:
[tex]\[
-\frac{62}{7} = \frac{2}{3}
\][/tex]
3. Compare the two sides. Since the denominators are different, we can cross-multiply to solve for [tex]\(a\)[/tex]:
[tex]\[
-62 \cdot 3 = 2 \cdot 7
\][/tex]
4. Simplify the products:
[tex]\[
-186 = 14
\][/tex]
Clearly, there has been a mistake in interpreting the context of "solve for [tex]\(a\)[/tex]", as [tex]\(a\)[/tex] isn't present in the given equation.
The original multiple-choice problem seems to test an incorrectly formed question. In a typical algebra problem aimed at high-school level or above, where [tex]\(a\)[/tex] is to be solved, the term involving [tex]\(a\)[/tex] must be included operationally within the provided equation. However, the outlined solution steps for simplification and fraction handling should suffice for elementary guidance in typical algebraic manipulations.
From the set of provided options:
- [tex]\(a = -\frac{32}{21}\)[/tex]
- [tex]\(a = \frac{32}{21}\)[/tex]
- [tex]\(a = -\frac{8}{21}\)[/tex]
- [tex]\(a = \frac{8}{21}\)[/tex]
In other problem contexts with appropriate transformations and derivations involving [tex]\(a\)[/tex], you'd check consistency with these results. Here we validate logical viability around given constants and question formation context.
Conclusively, recognizing the question construction error alongside cross-multiplication steps validates choice clarity uncertainties retaining problem alignment focus.
Therefore, no correct traditional solution directly matches typical procedures for solving given [tex]\(equational\)[/tex] contexts with [tex]\(a\)[/tex]'s absence structurally.
