To simplify the given expression [tex]\( \frac{2ax + 4bx}{3ay + 6by} \)[/tex], let's go through the steps in a detailed manner:
1. Identify Common Factors:
- In the numerator [tex]\( 2ax + 4bx \)[/tex], we notice that both terms [tex]\( 2ax \)[/tex] and [tex]\( 4bx \)[/tex] share a common factor of [tex]\(2x\)[/tex].
- In the denominator [tex]\( 3ay + 6by \)[/tex], we notice that both terms [tex]\( 3ay \)[/tex] and [tex]\( 6by \)[/tex] share a common factor of [tex]\(3y\)[/tex].
2. Factor Out the Common Terms:
- From the numerator [tex]\( 2ax + 4bx \)[/tex], factor out [tex]\(2x\)[/tex]:
[tex]\[
2ax + 4bx = 2x(a + 2b)
\][/tex]
- From the denominator [tex]\( 3ay + 6by \)[/tex], factor out [tex]\(3y\)[/tex]:
[tex]\[
3ay + 6by = 3y(a + 2b)
\][/tex]
3. Simplify the Expression:
- Substitute the factored forms back into the original fraction:
[tex]\[
\frac{2ax + 4bx}{3ay + 6by} = \frac{2x(a + 2b)}{3y(a + 2b)}
\][/tex]
- Notice that the term [tex]\((a + 2b)\)[/tex] appears in both the numerator and the denominator. Since [tex]\((a + 2b)\)[/tex] is a common factor, it can be cancelled out, provided [tex]\(a + 2b \neq 0\)[/tex]:
[tex]\[
\frac{2x(a + 2b)}{3y(a + 2b)} = \frac{2x}{3y}
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
\frac{2ax + 4bx}{3ay + 6by} = \frac{2x}{3y}
\][/tex]
