Problem 27 - Elementary Algebra

Question

For the following problems, add or subtract the rational expressions. $$ \frac{6 y}{5 x}+\frac{8 y}{5 x} $$

Step-by-Step Solution

Answer

Question: Add the following rational expressions: $$\frac{6y}{5x} + \frac{8y}{5x}$$ Answer: $$\frac{14y}{5x}$$

Step 1: Check if denominators are the same

In this case, the denominators are both "\(5x\)," so they are already the same.

Step 2: Add the numerators

Since the denominators are the same, you can add the numerators directly: $$ \frac{6y}{5x} + \frac{8y}{5x} $$

Step 3: Simplify the result

Add the numerators and place the result over the common denominator: $$ \frac{6y + 8y}{5x} $$

Step 4: Combine like terms

Combine the like terms in the numerator: $$ \frac{14y}{5x} $$ The result is: $$ \frac{14y}{5x} $$

Key Concepts

Common Denominator

When adding rational expressions, a critical first step is to have a common denominator. Why is this necessary? In essence, it's similar to ensuring everyone is 'speaking the same language' before starting a conversation. For our mathematical discourse, it means that we need to have the same base to combine our numbers effectively.

In our exercise, the rational expressions \frac{6y}{5x} and \frac{8y}{5x} already share a common denominator of \(5x\). This is akin to having two slices of cake that are the same size; it's easy to see how big the combined piece will be! If the denominators weren't the same, we would need to find equivalent fractions that share a common denominator before adding them together - just like slicing the cakes so that each slice pairs up perfectly with slices from the other.

Like Terms

After finding a common denominator, the next step in simplifying rational expressions is to combine like terms. In algebra, like terms are terms that have the same variable raised to the same power. Just think of them as members of the same family: they're similar enough that they can be grouped together.

In the provided expression \frac{6y}{5x} + \frac{8y}{5x}, both numerators, 6y and 8y, are like terms since they both contain the variable \(y\) to the first power. Combining 6y and 8y is straightforward - just add their coefficients (the number in front of the variable), giving us 14y. It's as simple as saying, 'If I have 6 apples and you give me 8 more, I now have 14 apples.'. The variable, like the type of fruit, doesn't change when combining.

Simplifying Expressions

The last key concept is simplifying expressions. This process is like cleaning up after cooking: you want the simplest, cleanest form possible so it's easy to understand what you've made. Simplification might involve combining like terms, reducing fractions, or factoring.

In our case, once we've added like terms to get \(14y\) in the numerator, we've actually already simplified the expression! There's no need to reduce or factor further. The expression \(\frac{14y}{5x}\) is tidy and concise. Simplifying is crucial because it helps us recognize when different expressions are actually the same, which is incredibly useful for solving equations and understanding relationships between variables.