To multiply the two fractions \(\frac{5x - 15}{7x + 21}\) and \(\frac{4x + 12}{14x - 42}\), we follow these steps:
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Factor each expression where possible.
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For \(5x - 15\): \[ 5x - 15 = 5(x - 3) \]
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For \(7x + 21\): \[ 7x + 21 = 7(x + 3) \]
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For \(4x + 12\): \[ 4x + 12 = 4(x + 3) \]
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For \(14x - 42\): \[ 14x - 42 = 14(x - 3) \]
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Substituting the factored forms into the fraction: \[ \frac{5(x - 3)}{7(x + 3)} \times \frac{4(x + 3)}{14(x - 3)} \]
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Multiply the fractions: \[ = \frac{5(x - 3) \cdot 4(x + 3)}{7(x + 3) \cdot 14(x - 3)} \]
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Cancel out the common factors:
- \(x - 3\) in the numerator and denominator.
- \(x + 3\) in the numerator and denominator.
After cancellation, we have: \[ = \frac{5 \cdot 4}{7 \cdot 14} = \frac{20}{98} \]
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Simplify \(\frac{20}{98}\):
- The greatest common divisor (GCD) of 20 and 98 is 2. \[ \frac{20 \div 2}{98 \div 2} = \frac{10}{49} \]
The final answer is: \[ \frac{10}{49} \]