To solve this problem, we need to determine which of the given options is equivalent to the equation [tex]\( x = 3y - 2 \)[/tex].
Let's start with each option:
1. [tex]\( x = y - \frac{11}{3} \)[/tex]
2. [tex]\( x = y + \frac{7}{3} \)[/tex]
3. [tex]\( x = 3\left(y - \frac{2}{3}\right) \)[/tex]
4. [tex]\( x = 3\left(y + \frac{2}{3}\right) \)[/tex]
Let's explore option 3 first:
Option 3: [tex]\( x = 3\left(y - \frac{2}{3}\right) \)[/tex]
We can distribute the 3 inside the parentheses to simplify the equation:
[tex]\[ x = 3 \cdot y - 3 \cdot \frac{2}{3} \][/tex]
[tex]\[ x = 3y - 2 \][/tex]
As we see, this simplifies to [tex]\( x = 3y - 2 \)[/tex], which matches the equation derived from solving for [tex]\( x \)[/tex].
Next, let's check the other options to be thorough:
Option 1: [tex]\( x = y - \frac{11}{3} \)[/tex]
There’s no way to manipulate [tex]\( x = y - \frac{11}{3} \)[/tex] to become [tex]\( x = 3y - 2 \)[/tex], as this option does not correctly scale or match the coefficient for [tex]\( y \)[/tex].
Option 2: [tex]\( x = y + \frac{7}{3} \)[/tex]
Similarly, [tex]\( x = y + \frac{7}{3} \)[/tex] cannot be adjusted to match [tex]\( x = 3y - 2 \)[/tex].
Option 4: [tex]\( x = 3\left(y + \frac{2}{3}\right) \)[/tex]
Distributing the 3 inside the parentheses results in:
[tex]\[ x = 3 \cdot y + 3 \cdot \frac{2}{3} \][/tex]
[tex]\[ x = 3y + 2 \][/tex]
This does not match the target equation [tex]\( x = 3y - 2 \)[/tex].
Therefore, the correct equivalent equation for [tex]\( x \)[/tex] derived from the original equation [tex]\( y = \frac{1}{3}(x+2) \)[/tex] is:
[tex]\[ x = 3\left(y - \frac{2}{3}\right) \][/tex]
So the correct answer is:
[tex]\[ \boxed{x = 3\left(y - \frac{2}{3}\right)} \][/tex]
