Question
5) \( \frac{4 a-5}{6 a^{2}+30 a}+\frac{a-1}{6 a^{2}+30 a} \)
Ask by Rogers Newton. in the United States
Jan 25,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\frac{5a - 6}{6a(a + 5)}
\]
Solution
To simplify the expression:
\[
\frac{4a - 5}{6a^{2} + 30a} + \frac{a - 1}{6a^{2} + 30a}
\]
**Step 1: Combine the Fractions**
Both fractions have the same denominator, so you can combine the numerators directly:
\[
\frac{4a - 5 + a - 1}{6a^{2} + 30a} = \frac{5a - 6}{6a^{2} + 30a}
\]
**Step 2: Factor the Denominator**
Factor out the greatest common factor (GCF) from the denominator:
\[
6a^{2} + 30a = 6a(a + 5)
\]
**Step 3: Rewrite the Expression with Factored Denominator**
\[
\frac{5a - 6}{6a(a + 5)}
\]
**Final Simplified Form:**
\[
\boxed{\dfrac{5a - 6}{6a(a + 5)}}
\]
This is the simplified form of the given expression.
Reviewed and approved by the UpStudy tutoring team
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Beyond the Answer
To combine the fractions, you can start by observing that both fractions have the same denominator: \[ \frac{4a - 5}{6a^2 + 30a} + \frac{a - 1}{6a^2 + 30a} = \frac{(4a - 5) + (a - 1)}{6a^2 + 30a} \] Now, simplify the numerator: \[ (4a - 5) + (a - 1) = 4a + a - 5 - 1 = 5a - 6 \] So, the expression simplifies to: \[ \frac{5a - 6}{6a^2 + 30a} \] Next, you can factor the denominator: \[ 6a^2 + 30a = 6a(a + 5) \] Now, the expression looks like this: \[ \frac{5a - 6}{6a(a + 5)} \] This is the simplified form of the given expression: \[ \frac{5a - 6}{6a(a + 5)} \]