Problem 42 - Elementary and Intermediate Algebra

Question

Solve each proportion. $$ \frac{8}{3 n+6}=\frac{16}{3 n-3} $$

Step-by-Step Solution

Answer

The solution to the proportion is \( n = -5 \).

Step 1: Cross-Multiply to Clear Fractions

To solve the proportion \( \frac{8}{3n + 6} = \frac{16}{3n - 3} \), we start by cross-multiplying. This means we'll multiply the numerator of each fraction by the denominator of the other fraction.This gives us:\[ 8(3n - 3) = 16(3n + 6) \]

Step 2: Distribute to Simplify the Equation

Now, apply the distributive property to both sides of the equation.Starting with the left side, distribute 8:\[ 8 \times 3n - 8 \times 3 = 24n - 24 \]For the right side, distribute 16:\[ 16 \times 3n + 16 \times 6 = 48n + 96 \]Now the equation is:\[ 24n - 24 = 48n + 96 \]

Step 3: Collect Like Terms

To solve for \( n \), we need to collect like terms by moving all terms involving \( n \) to one side and constant terms to the other.Subtract \( 24n \) from both sides to collect \( n \) terms:\[ 24n - 24 - 24n = 48n + 96 - 24n \]This simplifies to:\[ -24 = 24n + 96 \]

Step 4: Solve for n

Now isolate \( n \) by getting the terms with constants on one side of the equation.Subtract 96 from both sides:\[ -24 - 96 = 24n \]This simplifies to:\[ -120 = 24n \]Finally, divide both sides by 24 to solve for \( n \):\[ n = \frac{-120}{24} \]Simplify the fraction:\[ n = -5 \]

Step 5: Verify Your Solution

To ensure our solution is correct, substitute \( n = -5 \) back into the original proportion and see if both sides equal.Calculate \( 3n + 6 \) and \( 3n - 3 \) for \( n = -5 \):\[ 3(-5) + 6 = -15 + 6 = -9 \]\[ 3(-5) - 3 = -15 - 3 = -18 \]The original equation checks out as:Left side: \( \frac{8}{-9} \)Right side: \( \frac{16}{-18} = \frac{8}{-9} \)Both sides equal, confirming that \( n = -5 \) is indeed correct.

Key Concepts

Cross-Multiplication

Cross-multiplication is a fundamental technique that is useful when solving proportion problems. A proportion is an equation that shows two ratios or fractions are equal. To solve a proportion using cross-multiplication, you multiply the numerator of each fraction by the denominator of the other fraction.
For example, in the exercise, we had the proportion \( \frac{8}{3n + 6} = \frac{16}{3n - 3} \). By cross-multiplying, we multiply 8 with \(3n - 3\) and 16 with \(3n + 6\). So you end up with:

• \( 8(3n - 3) = 16(3n + 6) \)

This method "clears" the fractions, transforming the equation into a standard form that is easier to solve. Cross-multiplication is a handy method that simplifies the math and provides a straightforward way to start tackling a problem.

Distributive Property

The distributive property is another key concept when handling equations, as it allows you to simplify expressions. It states that distributing a single multiplier across terms inside a parenthesis is the same as multiplying each term individually by the said multiplier.
In our example:

• Distribute 8 into \(3n - 3\) to get \( 24n - 24 \).
• Distribute 16 into \(3n + 6\) to get \( 48n + 96 \).

This transformation helps you break down complex expressions into simpler, more manageable terms. It makes solving the equation more straightforward by clearing out parentheses and aligning terms properly for further simplification.

Solving Equations

After applying the distributive property, the resulting equation becomes more manageable: \( 24n - 24 = 48n + 96 \). The next step is to solve this equation by isolating the variable \( n \).
To do this, collect all terms involving \( n \) on one side of the equation and constants on the other:

• Subtract \( 24n \) from both sides to move all \( n \) terms together.
• Restore the balance by also taking constants to the opposite side — subtracting 96 from both sides.

This results in a simpler form \( -120 = 24n \), allowing us to easily find \( n \) by dividing both sides by 24, giving \( n = -5 \).
This step by step approach of collecting like terms and isolating the variable ensures clarity and precision, making sure you can solve the equation accurately.

Verifying Solutions

Verification is the final step in solving equations to ensure correctness. After solving for \( n \), it's essential to check if the solution makes the original proportion true.
Substitute \( n = -5 \) back into the original terms \( 3n+6 \) and \( 3n-3 \), calculating each:

• \( 3(-5) + 6 = -9 \)
• \( 3(-5) - 3 = -18 \)

Now confirm the original equation \( \frac{8}{-9} = \frac{16}{-18} \). The fractions on both sides simplify to the same value, \( \frac{8}{-9} \).
By verifying, you affirm that \( n = -5 \) is correct. Successful verification gives confidence that your calculations are correct and reinforces understanding of the equation.