Problem 5 - Algebra for College Students

Question

For Problems \(1-44\), solve each equation. $$ \frac{5}{n}+\frac{1}{3}=\frac{7}{n} $$

Step-by-Step Solution

Answer

The solution is \(n = 6\).

Step 1: Identify the Common Denominator

The common denominator of the fractions involved in the equation is the variable \(n\), as it is common to both \(\frac{5}{n}\) and \(\frac{7}{n}\). Thus, focus on clearing the fractions by using \(n\) as the common multiplier.

Step 2: Clear the Fractions

Multiply each term by \(n\) to eliminate the denominators. This gives: \((5) + \frac{n}{3} = 7\).

Step 3: Isolate Terms Involving Fractions

Subtract 5 from both sides to isolate the term involving the fraction: \(\frac{n}{3} = 7 - 5\), which simplifies to \(\frac{n}{3} = 2\).

Step 4: Solve for \(n\)

Multiply both sides of the equation by 3 to solve for \(n\). This results in: \(n = 2 \times 3\), which simplifies to \(n = 6\).

Key Concepts

Fraction Operations

When working with fractions in algebraic equations, understanding how to handle them is key. Here, fractions such as \( \frac{5}{n} \) and \( \frac{1}{3} \) require specific operations.

• Addition/Subtraction: To add or subtract fractions, they must have a common denominator. This allows you to 'combine' them seamlessly.
• Multiplication: Multiply the numerators and denominators separately. Useful when needing to "clear" fractions from an equation entirely.
• Inversion for Division: When dividing by a fraction, multiply by its reciprocal. This turns a division into a multiplication, making it easier to handle.

Recognizing these operations enables you to manipulate equations more effectively.

Solving Equations

Solving algebraic equations involves finding the value of the variable that makes the equation true. In our exercise, the goal was to solve \( \frac{5}{n} + \frac{1}{3} = \frac{7}{n} \).

• Balancing Equations: Perform the same operations on both sides to keep it balanced, akin to a scale.
• Simplification: Combine like terms or reduce fractions to make solving easier.
• Logical Steps: Approach the solution step-by-step, ensuring all manipulations are mathematically sound.

By systematically applying these principles, you can solve even complex equations.

Common Denominator

The common denominator in fractions is crucial when adding or subtracting them. In the problem, \( n \) is the common denominator for the fractions \( \frac{5}{n} \) and \( \frac{7}{n} \).

• Finding the Common Denominator: Identify a multiple common to all denominators involved.
• Rewriting Fractions: Once a common denominator is identified, rewrite each fraction with this denominator.
• Simplifying Equations: Allows fractions to be added or subtracted directly.

With a common denominator, you create a uniform base, facilitating easier equation manipulation.

Variable Isolation

Isolating the variable is often the end goal of solving an equation. This means you want the variable, like \( n \) in our exercise, all alone on one side.

• Step-by-Step Isolation: Use inverse operations to move other terms away from the variable.
• Transposing Terms: Shift terms not involving the variable to the other side of the equation.
• Fraction Clearing: Multiply to eliminate any fractions, as seen with \( \frac{n}{3} = 2 \), where multiplying by 3 gives \( n = 6 \).

By systematically isolating the variable, you can easily find its value.