To find an expression equivalent to \[ -\frac{2}{3} y - \frac{3}{5} x + \frac{2}{9} - \frac{4}{5} x + \frac{1}{3} y, \] we can first combine like terms.
Step 1: Combine the \( y \) terms
The coefficients of \( y \) in the expression are: \[ -\frac{2}{3} y + \frac{1}{3} y = \left(-\frac{2}{3} + \frac{1}{3}\right) y = -\frac{1}{3} y. \]
Step 2: Combine the \( x \) terms
The coefficients of \( x \) in the expression are: \[ -\frac{3}{5} x - \frac{4}{5} x = \left(-\frac{3}{5} - \frac{4}{5}\right) x = -\frac{7}{5} x. \]
Step 3: Combine the constant terms
The constant term in the expression is \(\frac{2}{9}\).
Step 4: Rewrite the expression
After combining everything, we have: \[ -\frac{1}{3} y - \frac{7}{5} x + \frac{2}{9}. \]
Alternative form
We can also express this as: \[ \frac{2}{9} - \frac{7}{5} x - \frac{1}{3} y. \]
Comparing with given options
Now let’s check each option for equivalence:
-
Option 1: \(y - \frac{4}{5} x + \frac{2}{9} - \frac{2}{3} y - \frac{1}{5} x\)
- This does not combine correctly to match our expression.
-
Option 2: \(-y - \frac{4}{5} x + \frac{2}{9} + \frac{2}{3} y + \frac{3}{5} x\)
- This does not match as it has \( y \) with different coefficients.
-
Option 3: \(\frac{1}{3} y - \frac{7}{5} x + \frac{2}{3} - \frac{2}{3} y - \frac{4}{9}\)
- This does not seem correct as we computed different coefficients.
-
Option 4: \( \frac{2}{3} y - \frac{7}{5} x + \frac{2}{9} + y\)
- This one also does not match our expression.
From the calculations, we can simplify our expression to: \[ -\frac{1}{3} y - \frac{7}{5} x + \frac{2}{9}. \] That does not appear directly in any option.
To summarize, the combination of terms simplified to this form gives us a correct equivalent expression, which was not listed.
Therefore, none of the options given are equivalent to the expression provided.