Which value of $n$ makes this equation true?
$\frac{3 n+3}{5}=\frac{5 n-1}{9}$
The problem asks us to solve the linear equation \(\frac{3n+3}{5} = \frac{5n-1}{9}\) for the variable \(n\). This equation is a proportion, where two ratios are set equal to each other. To solve for \(n\), we can clear the fractions to obtain a simpler linear equation.
To eliminate the denominators, we apply the property of cross-multiplication for proportions \(\frac{A}{B} = \frac{C}{D} \implies AD = BC\). Applying this to our equation gives:
\[9(3n + 3) = 5(5n - 1)\]
Next, we expand the expressions on both sides of the equation by distributing the constants:
\[27n + 27 = 25n - 5\]
Now we need to isolate the variable \(n\). First, we subtract \(25n\) from both sides of the equation:
\[27n - 25n + 27 = -5\]
\[2n + 27 = -5\]
Next, subtract 27 from both sides:
\[2n = -5 - 27\]
\[2n = -32\]
Finally, divide by 2:
\[n = \frac{-32}{2} = -16\]
We have found that \(n = -16\). We can verify this by substituting \(n = -16\) back into the original equation:
Left side: \(\frac{3(-16)+3}{5} = \frac{-48+3}{5} = \frac{-45}{5} = -9\).
Right side: \(\frac{5(-16)-1}{9} = \frac{-80-1}{9} = \frac{-81}{9} = -9\).
Since both sides are equal to -9, our solution is correct. The correct option is \(n = -16\).
This type of equation is essential in chemistry and physics, for instance, when balancing concentration ratios or finding the equivalent resistance in simple circuits. For example, if you are mixing a chemical solution in a specific ratio of parts, you might solve an equation like \(\frac{x+2}{3} = \frac{2x-1}{4}\) to find the exact amount of chemical needed, ensuring the reaction proceeds as expected. Mastering these linear equations is the foundation for solving more complex scientific and financial formulas.