Solve for z. 2/3 = 2 + 3/2 z – 1/2 z =

To solve for ( z ) in the equation:

[

frac{2}{3} = 2 + frac{3}{2} z – frac{1}{2}

]

First, simplify the right side of the equation:

1. Combine the constant terms ( 2 ) and ( -frac{1}{2} ):

[

2 – frac{1}{2} = frac{4}{2} – frac{1}{2} = frac{3}{2}

]

So the equation becomes:

[

frac{2}{3} = frac{3}{2} + frac{3}{2} z

]

2. Now, isolate the term with ( z ). Subtract ( frac{3}{2} ) from both sides:

[

frac{2}{3} – frac{3}{2} = frac{3}{2} z

]

3. To combine the fractions on the left, find a common denominator (which is 6):

[

frac{2}{3} = frac{4}{6}, quad frac{3}{2} = frac{9}{6}

]

So,

[

frac{4}{6} – frac{9}{6} = frac{-