Question

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question

simplify: \frac{6c}{5c + 7} + \frac{3}{5c + 7}.

provide your answer below:

Explanation:

Identify the common denominator

Using the Adding Rational Expressions knowledge point, we observe that both rational expressions already share a common denominator:

$$ 5c + 7 $$

Combine the numerators

Using the Adding Rational Expressions knowledge point, we add the numerators together over the shared denominator:

$$ \frac{6c}{5c + 7} + \frac{3}{5c + 7} = \frac{6c + 3}{5c + 7} $$

Factor the numerator

Using the Greatest Common Factor Factoring knowledge point, we factor out the greatest common factor, \(3\), from the numerator:

$$ 6c + 3 = 3(2c + 1) $$

Thus, the expression becomes:

$$ \frac{3(2c + 1)}{5c + 7} $$

Check for further simplification

We examine if any common factors exist between the numerator \(3(2c + 1)\) and the denominator \(5c + 7\). Since there are no common factors, the expression is in its simplest form:

$$ \frac{6c + 3}{5c + 7} \quad \text{or} \quad \frac{3(2c + 1)}{5c + 7} $$

Answer:

Simplify: <blank>\(\frac{6c+3}{5c+7}\)</blank>