To solve the equation [tex]\(\frac{2n}{3} = \frac{n+7}{2}\)[/tex], follow these steps:
1. Eliminate the fractions by finding a common multiple of the denominators.
- The denominators are 3 and 2. The least common multiple (LCM) of 3 and 2 is 6.
2. Multiply every term in the equation by this common multiple (6).
[tex]\[
6 \cdot \frac{2n}{3} = 6 \cdot \frac{n+7}{2}
\][/tex]
3. Simplify each term after multiplying:
- For the left side:
[tex]\[
6 \cdot \frac{2n}{3} = \left(6 \div 3\right) \times 2n = 2 \times 2n = 4n
\][/tex]
- For the right side:
[tex]\[
6 \cdot \frac{n + 7}{2} = \left(6 \div 2\right) \times (n + 7) = 3 \times (n + 7) = 3n + 21
\][/tex]
4. Rewrite the equation with these simplified terms:
[tex]\[
4n = 3n + 21
\][/tex]
5. Solve for [tex]\(n\)[/tex].
- First, isolate [tex]\(n\)[/tex] on one side of the equation by subtracting [tex]\(3n\)[/tex] from both sides:
[tex]\[
4n - 3n = 21
\][/tex]
- Simplify the left side:
[tex]\[
n = 21
\][/tex]
The value of [tex]\(n\)[/tex] is [tex]\(\boxed{21}\)[/tex].
