Let's solve the equation step by step:
Given: \[ 7(3a + 4) = 11(a - 1) + 19 \]
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Distribute on both sides: \[ 21a + 28 = 11a - 11 + 19 \]
Simplifying the right side: \[ 11a - 11 + 19 = 11a + 8 \] So we rewrite the equation as: \[ 21a + 28 = 11a + 8 \]
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Next, we'll move all terms involving \(a\) to one side and constant terms to the other side. Subtract \(11a\) from both sides: \[ 21a - 11a + 28 = 8 \] Simplifying gives: \[ 10a + 28 = 8 \]
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Now, subtract 28 from both sides: \[ 10a = 8 - 28 \] Simplifying gives: \[ 10a = -20 \]
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Finally, divide both sides by 10: \[ a = \frac{-20}{10} = -2 \]
Thus, the solution to the equation is: \[ \boxed{-2} \]