To determine which equation is equivalent to [tex]\( x = 3y - 2 \)[/tex], let's manipulate the options provided and see which one results in [tex]\( x = 3y - 2 \)[/tex].
1. Option: [tex]\( x = y - \frac{11}{3} \)[/tex]
Simplifying this, we have:
[tex]\[
x = y - \frac{11}{3}
\][/tex]
This equation does not seem to directly match [tex]\( x = 3y - 2 \)[/tex].
2. Option: [tex]\( x = y + \frac{7}{3} \)[/tex]
Simplifying this, we have:
[tex]\[
x = y + \frac{7}{3}
\][/tex]
This equation does not seem to directly match [tex]\( x = 3y - 2 \)[/tex].
3. Option: [tex]\( x = 3\left( y - \frac{2}{3} \right) \)[/tex]
Simplifying this, we handle the distribution:
[tex]\[
x = 3\left( y - \frac{2}{3} \right)
\][/tex]
Distribute the 3:
[tex]\[
x = 3y - 3 \cdot \frac{2}{3}
\][/tex]
[tex]\[
x = 3y - 2
\][/tex]
This is exactly the same as [tex]\( x = 3y - 2 \)[/tex].
4. Option: [tex]\( x = 3\left( y + \frac{2}{3} \right) \)[/tex]
Simplifying this, we handle the distribution:
[tex]\[
x = 3\left( y + \frac{2}{3} \right)
\][/tex]
Distribute the 3:
[tex]\[
x = 3y + 3 \cdot \frac{2}{3}
\][/tex]
[tex]\[
x = 3y + 2
\][/tex]
This equation does not match [tex]\( x = 3y - 2 \)[/tex].
Based on this simplification, the correct answer that is equivalent to [tex]\( x = 3y - 2 \)[/tex] is:
[tex]\[ x = 3\left( y - \frac{2}{3} \right) \][/tex]
Therefore, the equivalent equation is:
[tex]\[ x = 3\left( y - \frac{2}{3} \right) \][/tex]
Hence, the correct choice is [tex]\( x = 3\left(y - \frac{2}{3}\right) \)[/tex].
