To solve the equation [tex]\(\frac{3n + 3}{5} = \frac{5n - 1}{9}\)[/tex], we need to find the value of [tex]\(n\)[/tex] that satisfies this equation. Here's a detailed, step-by-step solution:
1. First, clear the fractions by finding a common denominator.
- The common denominator for 5 and 9 is 45.
- Multiply both sides of the equation by 45 to eliminate the denominators.
[tex]\[
\begin{aligned}
45 \cdot \frac{3n + 3}{5} & = 45 \cdot \frac{5n - 1}{9} \\
45 \cdot \frac{3n + 3}{5} & = 45 \cdot \frac{5n - 1}{9}
\end{aligned}
\][/tex]
2. Distribute the 45 to both terms inside each fraction:
[tex]\[
\begin{aligned}
45 \cdot \frac{3n + 3}{5} & = 45 \cdot \frac{5n - 1}{9} \\
\left(\frac{45}{5}\right) \cdot (3n + 3) & = \left(\frac{45}{9}\right) \cdot (5n - 1)
\end{aligned}
\][/tex]
3. Simplify the coefficients:
[tex]\[
\begin{aligned}
9(3n + 3) & = 5(5n - 1) \\
27n + 27 & = 25n - 5
\end{aligned}
\][/tex]
4. Move all terms involving [tex]\(n\)[/tex] to one side of the equation and constants to the other:
[tex]\[
\begin{aligned}
27n + 27 &= 25n - 5 \\
27n - 25n &= -5 - 27 \\
2n &= -32
\end{aligned}
\][/tex]
5. Solve for [tex]\(n\)[/tex]:
[tex]\[
\begin{aligned}
2n &= -32 \\
n &= \frac{-32}{2} \\
n &= -16
\end{aligned}
\][/tex]
6. Check the potential solutions among the given choices: [tex]\(n = -16\)[/tex], [tex]\(n = -2\)[/tex], [tex]\(n = 2\)[/tex], and [tex]\(n = 16\)[/tex]
- The correct answer is [tex]\(n = -16\)[/tex].
Therefore, the value of [tex]\(n\)[/tex] that makes the equation [tex]\(\frac{3n + 3}{5} = \frac{5n - 1}{9}\)[/tex] true is:
A. [tex]\(n = -16\)[/tex]
