Problem 27 - Elementary Algebra

Question

\(\frac{n}{49-n}=3+\frac{1}{49-n}\)

Step-by-Step Solution

Answer

The solution is \(n = 37\).

Step 1: Eliminate the Fraction

Start by multiplying both sides of the equation by the common denominator, which is \(49-n\). This gives:\[\frac{n}{49-n} imes (49-n) = (3 + \frac{1}{49-n}) imes (49-n)\]This simplifies to:\[n = 3(49-n) + 1\]

Step 2: Expand the Equation

Distribute the 3 on the right-hand side:\[n = 3 \times 49 - 3n + 1\]Simplifying further gives:\[n = 147 - 3n + 1\]Combine constants:\[n = 148 - 3n\]

Step 3: Rearrange to Isolate Variables

Add \(3n\) to both sides to bring all \(n\) terms to one side:\[n + 3n = 148\]Combine like terms:\[4n = 148\]

Step 4: Solve for n

Divide both sides by 4 to solve for \(n\):\[n = \frac{148}{4}\]Calculate the result:\[n = 37\]

Key Concepts

Common Denominator

When solving equations with fractions, a common denominator is your best friend. It allows you to simplify equations and get rid of the pesky fractional part. Imagine you see an equation with fractions that have the same denominator on both sides. By multiplying the entire equation by this common denominator, you can effectively clear the fractions. For example, in the equation \(\frac{n}{49-n} = 3 + \frac{1}{49-n}\), the common denominator is \(49-n\).

• Find the least common denominator for all fractional terms.
• Multiply every term in the equation by this number.
• Watch the fractions disappear, leaving you with a simpler equation.

This step sets a solid foundation for the remaining solution process, making the equation much more manageable.

Distributive Property

The distributive property is a fundamental algebraic principle that helps in expanding expressions. When you encounter a term multiplying a parenthesis, the distributive property becomes handy. To apply it, multiply the term outside the bracket by each term inside the bracket. In the equation \(n = 3(49 - n) + 1\), you need to distribute the 3 across both terms inside the parenthesis.

• Multiply the outer term (3) by each term inside the brackets (49 and \(-n\)).
• This gives \(3 \times 49 = 147\) and \(3 \times (-n) = -3n\).
• The expanded form becomes \(n = 147 - 3n + 1\).

Mastering the distributive property simplifies the complex steps of solving equations.

Combining Like Terms

Once the terms are distributed, the next step is combining like terms. This means adding or subtracting terms that have the same variable or are constants. For instance, in the equation \(n = 147 - 3n + 1\), combining like terms will clear up your expression.

• First, add the constants 147 and 1 to get 148.
• Recognize that like terms have the same letter (variable), so terms with 'n' can be combined.
• The equation becomes \(n = 148 - 3n\).

By combining like terms, the equation becomes simpler and gets you closer to the solution.

Isolating Variables

Isolating the variable is the final crucial step in solving equations. To find the value of the variable, you want to get it alone on one side of the equation. In our case, we are solving for \(n\), so we need to move all \(n\) terms to one side and constants to the other side.

• Add \(3n\) to both sides: \(n + 3n = 148\).
• This gives \(4n = 148\).
• Now, to isolate \(n\), divide both sides by 4.
• This results in \(n = \frac{148}{4}\), leading to \(n = 37\).

Isolating the variable is like peeling layers—each action brings you closer to the core, which is the solution.