To solve the equation \( \frac{2f}{5} - \frac{8}{5} + 2f = 8 \), we can follow these steps using the properties of equality:
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Combine like terms: Start with the terms that involve \( f \).
The term \( 2f \) can be written as \( \frac{2f}{1} \) and converted to have a common denominator with \( \frac{2f}{5} \). To do this, we can convert \( 2f \) into fifths: \[ 2f = \frac{10f}{5} \] Now, we can combine the terms involving \( f \): \[ \frac{2f}{5} + \frac{10f}{5} = \frac{12f}{5} \] Rewriting the original equation, we have: \[ \frac{12f}{5} - \frac{8}{5} = 8 \]
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Isolate the term with \( f \): To isolate \( \frac{12f}{5} \), we need to add \( \frac{8}{5} \) to both sides of the equation: \[ \frac{12f}{5} = 8 + \frac{8}{5} \]
To add \( 8 \) and \( \frac{8}{5} \), we will convert \( 8 \) into fifths: \[ 8 = \frac{40}{5} \] Thus, \[ 8 + \frac{8}{5} = \frac{40}{5} + \frac{8}{5} = \frac{48}{5} \]
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Now the equation is: \[ \frac{12f}{5} = \frac{48}{5} \]
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Eliminate the denominator: Multiply both sides of the equation by \( 5 \) to eliminate the fraction: \[ 12f = 48 \]
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Solve for \( f \): Divide both sides by \( 12 \): \[ f = \frac{48}{12} = 4 \]
Thus, the solution to the equation is: \[ \boxed{4} \]