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 final answer is − 16 .
Explanation
Problem Analysis 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. We can solve this equation by cross-multiplication to eliminate the fractions.
Eliminating Fractions Multiply both sides of the equation by 5 × 9 = 45 to eliminate the fractions: 45 × 5 3 n + 3 = 45 × 9 5 n − 1 This simplifies to: 9 ( 3 n + 3 ) = 5 ( 5 n − 1 )
Expanding the Equation Expand both sides of the equation: 27 n + 27 = 25 n − 5
Isolating n 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 Divide both sides by 2: 2 2 n = 2 − 32 n = − 16
Final Answer Therefore, the value of n that makes the equation true is n = − 16 .
Examples
In real-world scenarios, solving linear equations like this can help determine the break-even point in business. For example, if n represents the number of units you need to sell to make the revenue equal to the cost, this equation helps you find that specific number. Understanding how to manipulate and solve such equations is crucial for making informed decisions in various fields, including finance, economics, and engineering. By setting up equations that model real-world situations, we can find the values that optimize outcomes or meet specific criteria, such as balancing costs and revenues.
