Problem 13 - Algebra for College Students

Question

For Problems \(1-44\), solve each equation. $$ \frac{n}{65-n}=8+\frac{2}{65-n} $$

Step-by-Step Solution

Answer

The solution is \(n = 58\).

Step 1: Identify the Denominator

Notice that the common denominator in the equation is \(65-n\). This is important because we'll clear fractions by eliminating this common denominator.

Step 2: Clear the Fraction

Multiply both sides of the equation by \(65-n\) to eliminate the fractions: \(\frac{n}{65-n} \times (65-n) = \left(8+\frac{2}{65-n}\right) \times (65-n)\).

Step 3: Simplify the Equation

After multiplying, the equation becomes \(n = 8(65-n) + 2\). Expand the right side: \(n = 520 - 8n + 2\).

Step 4: Combine Like Terms

Reorder the equation to bring like terms together: \(n + 8n = 520 + 2\). Simplify to get \(9n = 522\).

Step 5: Solve for n

Divide both sides by 9 to isolate \(n\): \(n = \frac{522}{9}\). Simplify the division to find \(n = 58\).

Key Concepts

Fractions

Fractions are expressions representing parts of a whole. Often seen with a numerator and a denominator, they allow us to break down quantities into smaller, understandable units. In the original problem, we see fractions on both sides of the equation. Understanding fractions is key to solving fractional equations efficiently. The fraction \( \frac{n}{65-n} \) represents a part of \( 65-n \). Similarly, \( \frac{2}{65-n} \) is also a part of the same whole. By understanding fractions, you can navigate equations and simplify them by using the common denominator, which in this problem is \( 65-n \). This allows us to manipulate the equation to eventually isolate the variable and find its value.

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. It allows us to compare or combine fractions with different denominators by expressing them in terms of the same denominator. In our problem, both fractions share the common denominator \( 65-n \). By identifying this, it's possible to eliminate the fractions by multiplying each side of the equation by this common denominator. This strategy simplifies computations and enables us to proceed with solving the equation.

• Identify common denominators to simplify equations involving fractions.
• Eliminate fractions by multiplying through by the common denominator.

This manipulation makes it easier to solve the rest of the equation.

Like Terms

Like terms are terms in an equation that have the same variables raised to the same power. Recognizing and combining them is vital for simplifying equations. During the solution process, after expanding the equation, we combine the like terms to simplify:\[ n + 8n = 520 + 2 \]These are like terms as they both include the variable \( n \). By combining them, we turn the equation into a simpler form:

• Combine terms with the same variables to reduce the equation's complexity.

For our equation, adding \( n \) and \( 8n \) results in \( 9n \), helping us move closer to isolating the variable \( n \).

Isolating Variables

Isolating a variable means manipulating an equation to have the variable alone on one side. This strategy is used to identify the solution for that variable. Once the like terms are combined, the equation becomes simpler, as in:\[ 9n = 522 \]Here, our goal is to isolate \( n \). To do this, divide each side of the equation by 9, which gives:\[ n = \frac{522}{9} \]Finally, simplify the division to find \( n = 58 \).

• Rearrange the equation so that the variable is isolated on one side.
• Perform the necessary arithmetic operations to find the variable's value.

Isolating variables is a critical step in solving equations as it directly leads to determining the value of the unknown.