Select the correct answer.
Which value of $n$ makes this equation true?
$\frac{3 n+3}{5}=\frac{5 n-1}{9}$
A. $n=-16$
B. $n=-2$
C. $n=2$
D. $n=16$
Answer (1)
Multiply both sides by 45 to get rid of the fractions: 9 ( 3 n + 3 ) = 5 ( 5 n − 1 ) .
Expand both sides: 27 n + 27 = 25 n − 5 .
Simplify to isolate n : 2 n = − 32 .
Solve for n : n = − 16 . The correct answer is − 16 .
Explanation
Understanding the Problem We are given the equation 5 3 n + 3 = 9 5 n − 1 and asked to find the value of n that makes the equation true.
Eliminating Fractions To solve for n , we first eliminate the fractions by multiplying both sides of the equation by the least common multiple of 5 and 9, which is 45. This gives us: 45 × 5 3 n + 3 = 45 × 9 5 n − 1 9 ( 3 n + 3 ) = 5 ( 5 n − 1 )
Expanding the Equation Next, we expand both sides of the equation: 27 n + 27 = 25 n − 5
Isolating n Now, we want to isolate n on one side of the equation. Subtract 25 n from both sides: 27 n − 25 n + 27 = 25 n − 25 n − 5 2 n + 27 = − 5
Further Isolating n Subtract 27 from both sides: 2 n + 27 − 27 = − 5 − 27 2 n = − 32
Solving for n Finally, divide both sides by 2: 2 2 n = 2 − 32 n = − 16
Verification To verify our solution, we substitute n = − 16 back into the original equation: 5 3 ( − 16 ) + 3 = 9 5 ( − 16 ) − 1 5 − 48 + 3 = 9 − 80 − 1 5 − 45 = 9 − 81 − 9 = − 9 Since the equation holds true, our solution is correct.
Examples
In real-world scenarios, solving linear equations like this is crucial for determining unknown quantities in various fields. For instance, imagine you're comparing two different phone plans. Plan A costs $3 per month plus $0.05 per text, while Plan B costs $1 per month plus $0.09 per text. By setting up an equation similar to the one we solved, you can determine the number of texts you need to send for the two plans to cost the same. This helps you make an informed decision based on your usage habits, ensuring you choose the most cost-effective plan. The ability to solve such equations empowers you to make optimal choices in everyday situations involving costs, rates, and quantities.
