To solve the equation [tex]\(\frac{3n + 3}{5} = \frac{5n - 1}{9}\)[/tex], let's proceed step-by-step.
1. Start by eliminating the fractions. To do this, multiply both sides of the equation by the least common multiple (LCM) of 5 and 9, which is 45:
[tex]\[
45 \left(\frac{3n + 3}{5}\right) = 45 \left(\frac{5n - 1}{9}\right)
\][/tex]
2. Simplify both sides:
[tex]\[
9(3n + 3) = 5(5n - 1)
\][/tex]
3. Distribute the 9 and the 5 to get rid of the parentheses:
[tex]\[
27n + 27 = 25n - 5
\][/tex]
4. Move all the terms involving [tex]\(n\)[/tex] to one side of the equation by subtracting [tex]\(25n\)[/tex] from both sides:
[tex]\[
27n - 25n + 27 = -5
\][/tex]
Simplify:
[tex]\[
2n + 27 = -5
\][/tex]
5. Isolate [tex]\(n\)[/tex] by subtracting 27 from both sides:
[tex]\[
2n = -5 - 27
\][/tex]
Simplify:
[tex]\[
2n = -32
\][/tex]
6. Finally, solve for [tex]\(n\)[/tex] by dividing both sides by 2:
[tex]\[
n = \frac{-32}{2}
\][/tex]
Simplify:
[tex]\[
n = -16
\][/tex]
So, the value of [tex]\(n\)[/tex] that makes the equation true is [tex]\(\boxed{-16}\)[/tex].
