To solve for [tex]\( y \)[/tex] following the given steps, we have:
### 1. Distributive Property
First, apply the distributive property to the terms inside the parentheses:
[tex]\[
\frac{2}{5}\left(\frac{1}{2} y + 5\right) - \frac{4}{5} = \frac{1}{2} y - 1 + \frac{1}{10} y
\][/tex]
Distributing [tex]\( \frac{2}{5} \)[/tex] to both terms inside the parentheses:
[tex]\[
\frac{2}{5} \cdot \frac{1}{2} y + \frac{2}{5} \cdot 5 - \frac{4}{5} = \frac{1}{2} y - 1 + \frac{1}{10} y
\][/tex]
[tex]\[
\frac{1}{5} y + 2 - \frac{4}{5} = \frac{1}{2} y - 1 + \frac{1}{10} y
\][/tex]
### 2. Combine Like Terms
Next, combine like terms to simplify both sides:
[tex]\[
\frac{1}{5} y + 2 - \frac{4}{5} = \frac{1}{2} y - 1 + \frac{1}{10} y
\][/tex]
Combine constants on the left-hand side:
[tex]\[
\frac{1}{5} y + \frac{6}{5} = \frac{1}{2} y - 1 + \frac{1}{10} y
\][/tex]
Note that [tex]\( 2 - \frac{4}{5} = \frac{10}{5} - \frac{4}{5} = \frac{6}{5} \)[/tex].
### 3. Combine Terms on the Right-Hand Side
Combine the [tex]\( y \)[/tex]-terms on the right-hand side:
[tex]\[
\frac{1}{5} y + \frac{6}{5} = \frac{3}{5} y - 1
\][/tex]
Note that [tex]\( \frac{1}{2} y + \frac{1}{10} y = \frac{5}{10} y + \frac{1}{10} y = \frac{6}{10} y = \frac{3}{5} y \)[/tex].
Subtract [tex]\( \frac{1}{5} y \)[/tex] from both sides:
[tex]\[
\frac{6}{5} = \frac{3}{5} y - \frac{1}{5} y - 1
\][/tex]
[tex]\[
\frac{6}{5} = \frac{2}{5} y - 1
\][/tex]
### 4. Addition Property of Equality
Add 1 to both sides to isolate the term with [tex]\( y \)[/tex]:
[tex]\[
\frac{6}{5} + 1 = \frac{2}{5} y
\][/tex]
[tex]\[
\frac{6}{5} + \frac{5}{5} = \frac{2}{5} y
\][/tex]
[tex]\[
\frac{11}{5} = \frac{2}{5} y
\][/tex]
### 5. Division Property of Equality
Finally, solve for [tex]\( y \)[/tex] by dividing both sides by [tex]\(\frac{2}{5}\)[/tex]:
[tex]\[
y = \frac{\frac{11}{5}}{\frac{2}{5}}
\][/tex]
[tex]\[
y = \frac{11}{5} \cdot \frac{5}{2}
\][/tex]
[tex]\[
y = \frac{11 \cdot 5}{5 \cdot 2}
\][/tex]
[tex]\[
y = \frac{55}{10}
\][/tex]
[tex]\[
y = 5.5
\][/tex]
So, the value of [tex]\( y \)[/tex] is [tex]\( 5.5 \)[/tex].
