Let's solve the given problem step by step to determine which equation is equivalent to [tex]\( 4[x + 2(3x - 7)] = 22x - 65 \)[/tex].
1. First, simplify the left side of the equation [tex]\( 4[x + 2(3x - 7)] \)[/tex]:
[tex]\[
4[x + 2(3x - 7)]
\][/tex]
Start by simplifying inside the parenthesis:
[tex]\[
3x - 7
\][/tex]
Then multiply by 2:
[tex]\[
2(3x - 7) = 6x - 14
\][/tex]
Now, substitute back into the original expression:
[tex]\[
4[x + (6x - 14)] = 4[x + 6x - 14] = 4[7x - 14]
\][/tex]
Finally, distribute the 4:
[tex]\[
4(7x - 14) = 28x - 56
\][/tex]
2. Now, compare the simplified left side to the right side [tex]\( 22x - 65 \)[/tex]:
[tex]\[
28x - 56 = 22x - 65
\][/tex]
3. Next, let's look at the given list of equations and see which one matches [tex]\( 28x - 56 = 22x - 65 \)[/tex]:
Equation 1:
[tex]\[
28x - 7 = 22x - 65
\][/tex]
To check if this is equivalent, we rearrange it:
[tex]\[
28x - 22x - 7 = -65 \rightarrow 6x - 7 \neq -65
\][/tex]
This does not match the simplified equation [tex]\( 28x - 56 = 22x - 65 \)[/tex].
Equation 2:
[tex]\[
28x - 56 = 22x - 65
\][/tex]
This equation exactly matches the simplified form! Therefore, it is equivalent to the original equation [tex]\( 4[x + 2(3x - 7)] = 22x - 65 \)[/tex].
Equation 3:
[tex]\[
10x - 14 = 22x - 65
\][/tex]
Rearranging:
[tex]\[
10x - 22x - 14 = -65 \rightarrow -12x - 14 = -65 \rightarrow -12x = -51 \rightarrow x = \frac{51}{12}
\][/tex]
This does not match the simplified equation.
Equation 4:
[tex]\[
16x - 28 = 22x - 65
\][/tex]
Rearranging:
[tex]\[
16x - 22x - 28 = -65 \rightarrow -6x - 28 = -65 \rightarrow -6x = -37 \rightarrow x = \frac{37}{6}
\][/tex]
This also does not match the simplified equation.
After carefully examining all the options, we find that Equation 2:
[tex]\[
28x - 56 = 22x - 65
\][/tex]
is the one that is equivalent to the original equation.
Thus, the correct answer is:
[tex]\[ \boxed{2} \][/tex]
