Problem 1 - Introductory Algebra for College Students

Question

add or subtract as indicated. Simplify the result, if possible. $$\frac{7 x}{13}+\frac{2 x}{13}$$

Step-by-Step Solution

Answer

The answer is \( \frac{9x}{13} \).

Step 1: Identify common denominator

In this problem, the fractions \(\frac{7x}{13}+\frac{2x}{13}\) both have the same denominator, which is 13.

Step 2: Add the numerators

Keep the denominator as it is and add the numerators. \(7x + 2x\) = \(9x\).

Step 3: Write the result

The resultant fraction becomes \( \frac{9x}{13} \).

Key Concepts

Common Denominator

When simplifying algebraic fractions or combining fractions, one fundamental concept is finding a common denominator. This is a shared multiple of the denominators of two or more fractions that allows them to be added or subtracted. Having a common denominator is key because it ensures that the fractions are speaking the same 'mathematical language', so to speak, and can be directly compared or combined.

In our example, \(\frac{7x}{13}+\frac{2x}{13}\), both fractions have the same denominator, 13. This means they already share a common denominator, which makes the process of combining them straightforward. The process becomes more challenging when dealing with different denominators, as they require a step to find an equivalent fraction for each that has the same denominator. Remember, the common denominator should be the least common multiple (LCM) of the denominators for the most simplified solution.

Addition of Fractions

Once you have a common denominator, the addition of fractions is quite simple. You keep the denominator the same and only add the numerators. Think of it as having a pie cut into equal slices; if you want to know how much of the pie you have in total, you count the slices (the numerators), not alter the size of each slice (the denominator).

Using our initial exercise, since we have \(\frac{7x}{13}+\frac{2x}{13}\), we simply add the numerators (7x and 2x) to get 9x, while the denominator remains unchanged. Therefore, the sum of these fractions is \(\frac{9x}{13}\). It's important to always look for opportunities to simplify the resulting fraction, but in this case, 9x and 13 have no common factors other than 1, so the fraction is already in its simplest form.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operations (like addition and multiplication). In algebra, we often work with expressions in the form of fractions. These algebraic fractions can represent parts of wholes, ratios, or proportions and follow the same rules for arithmetic as ordinary fractions.

Our original problem, \(\frac{7x}{13}+\frac{2x}{13}\), involves adding algebraic fractions. The variable 'x' is a placeholder that represents a number we don't yet know. When simplifying algebraic expressions, it's important to combine like terms (terms that have the same variable raised to the same power) and apply the order of operations properly. In this case, we only dealt with like terms (7x and 2x), which made the addition straightforward, resulting in a simplified algebraic fraction of \(\frac{9x}{13}\). In more complex expressions, you may need to distribute, factor, or use other algebraic techniques to combine or simplify.