To determine which of the given options is an ordered pair of the relation \( y = -\frac{2}{3}x + 6 \), we can plug in the \( x \) values from each option and see if the resulting \( y \) value matches the provided \( y \) value in the ordered pair.
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For \( (1, 4) \): \[ y = -\frac{2}{3}(1) + 6 = -\frac{2}{3} + 6 = -\frac{2}{3} + \frac{18}{3} = \frac{16}{3} \neq 4 \]
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For \( (0, -\frac{2}{3}) \): \[ y = -\frac{2}{3}(0) + 6 = 0 + 6 = 6 \neq -\frac{2}{3} \]
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For \( (3, 8) \): \[ y = -\frac{2}{3}(3) + 6 = -2 + 6 = 4 \neq 8 \]
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For \( (3, 4) \): \[ y = -\frac{2}{3}(3) + 6 = -2 + 6 = 4 = 4 \]
The only ordered pair that satisfies the equation \( y = -\frac{2}{3}x + 6 \) is \( (3, 4) \).
Thus, the correct answer is \( (3, 4) \).