To solve for [tex]\( x \)[/tex] from the given equation [tex]\( y = \frac{1}{3}(x + 2) \)[/tex], we'll perform a series of algebraic steps. Let's go through it step by step:
1. Start with the given equation:
[tex]\[
y = \frac{1}{3}(x + 2)
\][/tex]
2. To eliminate the fraction, multiply both sides by 3:
[tex]\[
3y = x + 2
\][/tex]
3. Next, isolate [tex]\( x \)[/tex] by subtracting 2 from both sides:
[tex]\[
x = 3y - 2
\][/tex]
Now that we have [tex]\( x = 3y - 2 \)[/tex], let's compare it with the given options to find the equivalent equation:
1. Option 1: [tex]\( x = y - \frac{11}{3} \)[/tex]
[tex]\[
x = y - \frac{11}{3}
\][/tex]
This does not match our equation.
2. Option 2: [tex]\( x = y + \frac{7}{3} \)[/tex]
[tex]\[
x = y + \frac{7}{3}
\][/tex]
This does not match our equation.
3. Option 3: [tex]\( x = 3\left(y - \frac{2}{3}\right) \)[/tex]
[tex]\[
x = 3 \left( y - \frac{2}{3} \right)
\][/tex]
Simplifying the right-hand side:
[tex]\[
x = 3y - 2
\][/tex]
This matches our derived equation [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]
Simplifying the right-hand side:
[tex]\[
x = 3y + 2
\][/tex]
This does not match our equation.
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].
