To simplify the expression \( \frac{7}{9} h + 10 - \frac{4}{9} h - 3 \), let's combine like terms.
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Combine the \( h \) terms: \[ \frac{7}{9} h - \frac{4}{9} h = \left(\frac{7}{9} - \frac{4}{9}\right) h = \frac{3}{9} h = \frac{1}{3} h \]
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Combine the constant terms: \[ 10 - 3 = 7 \]
Putting it all together, we get: \[ \frac{1}{3} h + 7 \]
Thus, the simplified expression is: \[ \frac{1}{3} h + 7 \]