To find the value of [tex]\( y \)[/tex] that satisfies the equations [tex]\( x = y + 2 \)[/tex] and [tex]\( y - \frac{x - 2}{2} = \frac{2}{3} \)[/tex], we can follow these steps:
1. Substitute [tex]\( x \)[/tex] in the equation:
We know that [tex]\( x = y + 2 \)[/tex]. Substitute this expression for [tex]\( x \)[/tex] in the given equation [tex]\( y - \frac{x - 2}{2} = \frac{2}{3} \)[/tex].
[tex]\[
y - \frac{(y + 2 - 2)}{2} = \frac{2}{3}
\][/tex]
2. Simplify the inner expression:
Simplify the term inside the fraction:
[tex]\[
y - \frac{y + 2 - 2}{2} = y - \frac{y + 0}{2} = y - \frac{y}{2}
\][/tex]
3. Combine like terms:
Now, combine the like terms in the equation:
[tex]\[
y - \frac{y}{2} = \frac{2}{3}
\][/tex]
4. Simplify the expression:
Simplify [tex]\( y - \frac{y}{2} \)[/tex]. Note that [tex]\( y \)[/tex] can be written as [tex]\( \frac{2y}{2} \)[/tex]:
[tex]\[
\frac{2y}{2} - \frac{y}{2} = \frac{2y - y}{2} = \frac{y}{2}
\][/tex]
Now our equation looks like this:
[tex]\[
\frac{y}{2} = \frac{2}{3}
\][/tex]
5. Solve for [tex]\( y \)[/tex]:
Finally, solve the equation [tex]\( \frac{y}{2} = \frac{2}{3} \)[/tex]. To isolate [tex]\( y \)[/tex], multiply both sides of the equation by 2:
[tex]\[
y = 2 \cdot \frac{2}{3}
\][/tex]
[tex]\[
y = \frac{4}{3}
\][/tex]
So, the value of [tex]\( y \)[/tex] that satisfies the given equation is [tex]\( \boxed{\frac{4}{3}} \)[/tex].
