To determine which of the provided options is equivalent to the equation [tex]\( x = 3y - 2 \)[/tex], let's analyze each option step-by-step:
1. Option 1: [tex]\( x = y - \frac{11}{3} \)[/tex]
[tex]\[
x = y - \frac{11}{3}
\][/tex]
Let's rewrite this in another form to compare it with [tex]\( x = 3y - 2 \)[/tex]:
[tex]\[
x = y - \frac{11}{3} = \frac{3y}{3} - \frac{11}{3} = \frac{3y - 11}{3}
\][/tex]
This is not equivalent to [tex]\( x = 3y - 2 \)[/tex].
2. Option 2: [tex]\( x = y + \frac{7}{3} \)[/tex]
[tex]\[
x = y + \frac{7}{3}
\][/tex]
Let's rewrite this in another form to compare it with [tex]\( x = 3y - 2 \)[/tex]:
[tex]\[
x = y + \frac{7}{3} = \frac{3y}{3} + \frac{7}{3} = \frac{3y + 7}{3}
\][/tex]
This is not equivalent to [tex]\( x = 3y - 2 \)[/tex].
3. Option 3: [tex]\( x = 3\left(y - \frac{2}{3}\right) \)[/tex]
[tex]\[
x = 3\left(y - \frac{2}{3}\right)
\][/tex]
Let's simplify this:
[tex]\[
x = 3\left(y - \frac{2}{3}\right) = 3y - 3 \cdot \frac{2}{3} = 3y - 2
\][/tex]
This is exactly equivalent to [tex]\( x = 3y - 2 \)[/tex].
4. Option 4: [tex]\( x = 3\left(y + \frac{2}{3}\right) \)[/tex]
[tex]\[
x = 3\left(y + \frac{2}{3}\right)
\][/tex]
Let's simplify this:
[tex]\[
x = 3\left(y + \frac{2}{3}\right) = 3y + 3 \cdot \frac{2}{3} = 3y + 2
\][/tex]
This is not equivalent to [tex]\( x = 3y - 2 \)[/tex].
After analyzing all the options:
- Option 1 is not equivalent.
- Option 2 is not equivalent.
- Option 3 is equivalent.
- Option 4 is not equivalent.
Therefore, the equivalent equation for [tex]\( x \)[/tex] is:
[tex]\[ x = 3\left(y - \frac{2}{3}\right) \][/tex]
Thus, the correct option is:
[tex]\[ \boxed{3} \][/tex]
