To solve the equation [tex]\(2 + 3(2a + 1) = 3(a + 2)\)[/tex] step-by-step, follow these steps:
1. Distribute the constants on both sides:
The left-hand side starts as [tex]\(2 + 3(2a + 1)\)[/tex]. Distribute the 3 within the parentheses:
[tex]\[
2 + 3 \cdot 2a + 3 \cdot 1
\][/tex]
This simplifies to:
[tex]\[
2 + 6a + 3
\][/tex]
The right-hand side is [tex]\(3(a + 2)\)[/tex]. Distribute the 3:
[tex]\[
3 \cdot a + 3 \cdot 2
\][/tex]
This simplifies to:
[tex]\[
3a + 6
\][/tex]
2. Combine like terms:
On the left-hand side [tex]\(2 + 6a + 3\)[/tex], combine the constants 2 and 3:
[tex]\[
5 + 6a
\][/tex]
So the equation now is:
[tex]\[
5 + 6a = 3a + 6
\][/tex]
3. Move all terms involving [tex]\(a\)[/tex] to one side of the equation and constants to the other:
Subtract [tex]\(3a\)[/tex] from both sides to get the terms involving [tex]\(a\)[/tex] on one side:
[tex]\[
6a - 3a = 6 - 5
\][/tex]
This simplifies to:
[tex]\[
3a = 1
\][/tex]
4. Solve for [tex]\(a\)[/tex]:
Divide both sides by 3 to isolate [tex]\(a\)[/tex]:
[tex]\[
a = \frac{1}{3}
\][/tex]
Thus, the solution to the equation [tex]\(2 + 3(2a + 1) = 3(a + 2)\)[/tex] is [tex]\( a = \frac{1}{3} \)[/tex].
Among the given choices:
1) [tex]\(\frac{1}{7}\)[/tex]
2) [tex]\(\frac{1}{3}\)[/tex]
3) [tex]\(-\frac{3}{7}\)[/tex]
4) [tex]\(-\frac{1}{3}\)[/tex]
The correct answer is choice 2) [tex]\(\frac{1}{3}\)[/tex].
