Problem 16 - Prealgebra and Introductory Algebra

Question

Simplify. $$\frac{2 n}{3 n+4}-\frac{5 n-3}{3 n+4}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{-3n + 3}{3n + 4}\).

Step 1: Identify common denominator

The fractions have same denominator, which is \(3n + 4\). So that's our common denominator.

Step 2: Subtract numerators

Now, subtract the numerators. Doing so, \((2n) - (5n - 3)\) gives us \(-3n + 3\). Remember to distribute the minus sign to both terms in the second numerator.

Step 3: Write the answer as a single fraction

Now we write this result over our common denominator. That gives us \(\frac{-3n + 3}{3n + 4}\) as our final result.

Key Concepts

Common Denominator

Dealing with algebraic fractions can be easier when you understand the role of the common denominator. A denominator is the bottom part of a fraction, and when fractions share the same denominator, they are much simpler to work with. Finding a common denominator allows you to perform operations such as addition, subtraction, multiplication, and division on fractions.

When two algebraic fractions share a common denominator, you can think of it like having a shared foundation. This shared base lets you directly compare or combine the numerators (the top numbers of the fractions) since the denominators do not change. In the exercise we're looking at, both fractions have the denominator \(3n + 4\), which neatly sets us up for the next steps without needing any further adjustments or modifications to the fractions themselves.

Subtracting Numerators

Once a common denominator is established, subtracting numerators becomes our focus. The numerator represents the number of parts we have out of the whole (denominator). When you subtract numerators of fractions with the same denominator, you're essentially determining how many parts remain or the difference between the parts.

In the context of algebraic fractions, it is crucial to carefully subtract the numerators. Remember that subtracting a negative is like adding a positive. This can be seen in the given problem where we subtract \(5n - 3\) from \(2n\), making it important to distribute the minus sign to both terms. As a result, we get \(2n - 5n + 3\), which simplifies to \(–3n + 3\). The act of combining like terms is also part of this step, and we must be attentive to signs and coefficients.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (such as addition and subtraction). In this exercise, we're working with algebraic expressions within fractions. Manipulating these expressions correctly is paramount to arriving at the correct simplified form.

The key is to treat each part of the expression with care, particularly when distributing signs or combining like terms. You need to ensure that you do not change the value of the expression, only its form. This concept is about maintaining the 'equivalence' as you transform and simplify the expression. In our problem, after subtracting the numerators wisely and keeping the common denominator, we obtain the simplified algebraic fraction: \(\frac{-3n + 3}{3n + 4}\). Maintaining the integrity of the algebraic expression while simplifying allows for a clear and correct solution to be presented.