To solve the equation \(27x + 8 = -2\frac{2}{7}x + 8\), we first need to convert the mixed number \(-2\frac{2}{7}\) into an improper fraction.
\[ -2\frac{2}{7} = -\left(2 \times 7 + 2\right)/7 = -\frac{14 + 2}{7} = -\frac{16}{7} \]
Now we can rewrite the equation as:
\[ 27x + 8 = -\frac{16}{7}x + 8 \]
Next, we'll eliminate the \(8\) from both sides of the equation:
\[ 27x = -\frac{16}{7}x \]
Now, let's isolate \(x\). To do this, we can add \(\frac{16}{7}x\) to both sides:
\[ 27x + \frac{16}{7}x = 0 \]
To combine these two terms, we need a common denominator. The common denominator between \(27\) and \(7\) is \(189\). Rewriting \(27x\) with this common denominator gives:
\[ 27x = \frac{27 \times 7}{7}x = \frac{189}{7}x \]
Now we have:
\[ \frac{189}{7}x + \frac{16}{7}x = 0 \]
Combine the left-hand side:
\[ \frac{189 + 16}{7}x = 0 \implies \frac{205}{7}x = 0 \]
Now, for the equation to hold true, since \(\frac{205}{7} \neq 0\), we need:
\[ x = 0 \]
Thus, the solution to the equation \(27x + 8 = -2\frac{2}{7}x + 8\) is:
\[ \boxed{0} \]
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