To express the given expression as a single fraction and simplify it, let’s proceed step-by-step. The initial expression is:
[tex]\[
8 + \frac{a - 7b}{3a} - \frac{8a + 5b}{2a}
\][/tex]
### Step 1: Combine into a single fraction with a common denominator
Let's first identify the individual fractions and their denominators:
- The first term [tex]\(8\)[/tex] can be written as [tex]\(\frac{8 \cdot 3a \cdot 2a}{3a \cdot 2a}\)[/tex] to match the common denominator.
- The second term already has the denominator [tex]\(3a\)[/tex].
- The third term has the denominator [tex]\(2a\)[/tex].
Find the least common multiple (LCM) of the denominators [tex]\(3a\)[/tex] and [tex]\(2a\)[/tex], which is [tex]\(6a\)[/tex].
### Step 2: Rewrite each term with the common denominator [tex]\(6a\)[/tex]
Rewrite the given expression with the common denominator:
1. Convert [tex]\(8\)[/tex] to have the denominator [tex]\(6a\)[/tex]:
[tex]\[
8 = \frac{8 \cdot 6a}{6a} = \frac{48a}{6a}
\][/tex]
2. Convert [tex]\(\frac{a - 7b}{3a}\)[/tex] to have the denominator [tex]\(6a\)[/tex]:
[tex]\[
\frac{a - 7b}{3a} = \frac{2(a - 7b)}{6a} = \frac{2a - 14b}{6a}
\][/tex]
3. Convert [tex]\(\frac{8a + 5b}{2a}\)[/tex] to have the denominator [tex]\(6a\)[/tex]:
[tex]\[
\frac{8a + 5b}{2a} = \frac{3(8a + 5b)}{6a} = \frac{24a + 15b}{6a}
\][/tex]
### Step 3: Combine all fractions over the common denominator [tex]\(6a\)[/tex]
Now, add and subtract the fractions:
[tex]\[
\frac{48a}{6a} + \frac{2a - 14b}{6a} - \frac{24a + 15b}{6a}
\][/tex]
Combine the numerators:
[tex]\[
\frac{48a + (2a - 14b) - (24a + 15b)}{6a}
\][/tex]
### Step 4: Simplify the numerator
Combine and simplify the terms in the numerator:
[tex]\[
48a + 2a - 14b - 24a - 15b = (48a + 2a - 24a) + (-14b - 15b) = 26a - 29b
\][/tex]
### Step 5: Write the simplified fraction
The simplified fraction is:
[tex]\[
\frac{26a - 29b}{6a}
\][/tex]
Thus, the expression [tex]\(8 + \frac{a - 7b}{3a} - \frac{8a + 5b}{2a}\)[/tex] simplifies to:
[tex]\[
\frac{26a - 29b}{6a}
\][/tex]
