Let's simplify the given expression step-by-step:
[tex]\[
2 + \frac{9a - 4x}{6a} - \frac{5a - x}{2a}
\][/tex]
### Step 1: Express all terms with a common denominator.
The first term is a whole number. To combine it with the fractions, we convert it to a fraction with the same common denominator.
We start by identifying a common denominator for the fractions [tex]\(\frac{9a - 4x}{6a}\)[/tex] and [tex]\(\frac{5a - x}{2a}\)[/tex].
The denominators are [tex]\(6a\)[/tex] and [tex]\(2a\)[/tex]. The least common multiple (LCM) of these denominators is [tex]\(6a\)[/tex]. Therefore, we will convert each fraction's denominator to [tex]\(6a\)[/tex].
#### Convert [tex]\(\frac{5a - x}{2a}\)[/tex] to [tex]\(\frac{k}{6a}\)[/tex]
To make the denominator of [tex]\(\frac{5a - x}{2a}\)[/tex] equal to [tex]\(6a\)[/tex], we multiply both the numerator and the denominator by 3:
[tex]\[
\frac{5a - x}{2a} \cdot \frac{3}{3} = \frac{3(5a - x)}{6a} = \frac{15a - 3x}{6a}
\][/tex]
#### Express [tex]\(2\)[/tex] as a fraction with denominator [tex]\(6a\)[/tex]
To express [tex]\(2\)[/tex] with denominator [tex]\(6a\)[/tex], we multiply it by [tex]\(\frac{6a}{6a}\)[/tex]:
[tex]\[
2 = 2 \cdot \frac{6a}{6a} = \frac{12a}{6a}
\][/tex]
Now we rewrite the original expression with a common denominator [tex]\(6a\)[/tex]:
[tex]\[
\frac{12a}{6a} + \frac{9a - 4x}{6a} - \frac{15a - 3x}{6a}
\][/tex]
### Step 2: Combine the fractions
Since all terms have the common denominator [tex]\(6a\)[/tex], we can combine the numerators directly:
[tex]\[
\frac{12a + (9a - 4x) - (15a - 3x)}{6a}
\][/tex]
### Step 3: Simplify the numerator
Distribute the negative sign in the second part of the numerator:
[tex]\[
= \frac{12a + 9a - 4x - 15a + 3x}{6a}
\][/tex]
Combine like terms in the numerator:
[tex]\[
= \frac{12a + 9a - 15a + (-4x + 3x)}{6a} = \frac{(12a + 9a - 15a) + (-4x + 3x)}{6a}
\][/tex]
[tex]\[
= \frac{(12a - 6a) - x}{6a} = \frac{6a - x}{6a}
\][/tex]
### Step 4: Simplify the fraction
We can simplify the numerator and denominator by canceling the common factor [tex]\(a\)[/tex]:
[tex]\[
= \frac{6a - x}{6a}
\][/tex]
Thus, the simplified answer is:
[tex]\[
\frac{6a - x}{6a}
\][/tex]
This is the simplified single fraction representation of the given expression.
