Problem 12 - Prealgebra and Introductory Algebra

Question

Simplify. $$\frac{4 x+3}{2 x-7}+\frac{3 x-8}{2 x-7}$$

Step-by-Step Solution

Answer

\(\frac{7x - 5}{2x - 7}\)

Step 1: Identifying the Denominators

The fractions have the same denominator, which is \(2x - 7\). In such a case we can easily add the numerators while keeping the denominator same.

Step 2: Adding the Numerators

The numerators of the fractions are \(4x + 3\) and \(3x - 8\). Adding these two together will give a new numerator. So, the resulting numerator will be \((4x + 3) + (3x - 8) = 7x - 5\).

Step 3: Forming the New Fraction

Now that the new numerator is \(7x - 5\), you can form the new simplified fraction, keeping the denominator the same. The new fraction will be \(\frac{7x - 5}{2x - 7}\).

Key Concepts

Simplifying Expressions

Simplifying algebraic expressions involving fractions can sound intimidating, but it's often just about rearranging and combining terms to make the expression more straightforward. In this case, we aim to simplify the sum of two fractions. The key steps involve treating the expression's components—numerators and denominators alike—with care and precision.
We first determine if the denominators are the same. If they are, this significantly simplifies the process, as the expression can then be condensed by directly combining their numerators. This essentially turns a more complex problem into a basic arithmetic exercise involving addition.
When simplifying expressions in algebra, always remember:

• Combine like terms where possible;
• Factorize complex numerators or use expansion where needed;
• Reduce fractions if any common factors appear.

Once simplified, the fraction represents a clearer version of the original expression, easier to read and understand.

Common Denominator

In the world of fractions, a common denominator is your best friend. It's what allows two fractions with different numerators to be added or subtracted seamlessly. In this exercise, both fractions already share the common denominator of \(2x - 7\), saving us from additional steps, such as finding an equivalent denominator for both fractions.
Why is having a common denominator important? Imagine calculating sums with two different units—it's confusing, isn't it? That's what fractions with different denominators seem like. By ensuring the bottom part of the fraction (the denominator) matches, you streamline the process and directly work with the numerators.
When different terms share the same denominator, you can:

• Add or subtract numerators directly;
• Focus purely on the algebraic manipulation of numerators;
• Rest assured that your simplified expression reflects the same original relationships.

So, a common denominator is like translating different languages into one understandable format.

Adding Fractions

Adding fractions in algebra is straightforward once you have ensured a common denominator. This exercise exemplifies that simplicity. With both fractions sharing the denominator \(2x - 7\), we can proceed by simply adding the numerators: \(4x + 3\) and \(3x - 8\).
Let's break it down further to avoid any hurdles:

• Keep the common denominator as it is – no changes needed there;
• Add the numerators together: \((4x + 3) + (3x - 8)\) gives us \(7x - 5\);
• Form a new fraction from the summed numerator and the unchanged common denominator: \(\frac{7x - 5}{2x - 7}\).

Remember, adding fractions is just combining numerators when denominators match. It simplifies what could otherwise be a complex operation to a single, clear expression. By following the basic rules, you streamline problem-solving in algebra.