Question

(y + 5x)\frac{1}{2}
a(8 + 2b - 6)

Explanation:

First Expression: \((y + 5x)\frac{1}{2}\)

Step1: Distribute the \(\frac{1}{2}\)

To simplify \((y + 5x)\frac{1}{2}\), we use the distributive property \(a(b + c)=ab+ac\). Here \(a = \frac{1}{2}\), \(b = y\), and \(c = 5x\).
\[
(y + 5x)\frac{1}{2}=\frac{1}{2}y+\frac{1}{2}(5x)
\]

Step2: Simplify the terms

Simplify \(\frac{1}{2}(5x)\) to \(\frac{5}{2}x\) and \(\frac{1}{2}y\) can be written as \(\frac{y}{2}\). So the simplified form is:
\[
\frac{y}{2}+\frac{5x}{2}
\]
We can also factor out \(\frac{1}{2}\) to get \(\frac{1}{2}(y + 5x)\) (which is the original form, but if we want to expand, we get \(\frac{y + 5x}{2}\))

Second Expression: \(a(8 + 2b - 6)\)

Step1: Combine like terms inside the parentheses

First, combine the constant terms \(8-6\) inside the parentheses.
\[
a(8 + 2b - 6)=a((8 - 6)+2b)
\]
\[
=a(2 + 2b)
\]

Step2: Distribute the \(a\)

Now use the distributive property \(a(b + c)=ab+ac\) where \(a=a\), \(b = 2\), and \(c = 2b\).
\[
a(2 + 2b)=2a+2ab
\]
We can also factor out a 2 from the terms inside the parentheses first: \(a(2(1 + b)) = 2a(1 + b)\) (which is equivalent to \(2a+2ab\))

Answer:

s:
For \((y + 5x)\frac{1}{2}\): \(\frac{y + 5x}{2}\) (or \(\frac{y}{2}+\frac{5x}{2}\))
For \(a(8 + 2b - 6)\): \(2a + 2ab\) (or \(2a(1 + b)\))