Problem 62 - Elementary Algebra

Question

For the following problems, perform the indicated operations. $$ \frac{7 y+4}{6 y^{2}-32 y+32}+\frac{6 y-10}{2 y^{2}-18 y+40} $$

Step-by-Step Solution

Answer

Question: Add the following rational expressions and simplify if possible: $$\frac{7y + 4}{6y^2 - 32y + 32} + \frac{6y - 10}{2y^2 - 18y + 40}$$ Answer: $$\frac{10y - 1}{y^2 - 9y + 20}$$

Step 1: Identify the denominators in given expressions

In our given expressions, we have two denominators: $$ 6y^2 - 32y + 32 $$ and $$ 2y^2 - 18y + 40 $$

Step 2: Find the least common denominator (LCD)

To find the least common denominator, we need to find the least common multiple of the two denominators. Notice that the second denominator is just two times the first one, which means that the least common multiple is the second denominator: $$ \text{LCD} = 2y^2 -18y + 40 $$

Step 3: Convert both expressions to have the same denominator

Now, we need to modify both expressions to make their denominators equal to the LCD. For the first expression, we will multiply both the numerator and denominator by 2: $$ \frac{7y + 4}{6y^2 - 32y + 32} \times \frac{2}{2} = \frac{14y + 8}{2y^2 - 18y + 40} $$ The second expression already has the LCD as its denominator, so there's no need to modify it.

Step 4: Add the modified expressions

Now that both expressions have the same denominator, we can add them by combining their numerators: $$ \frac{14y + 8}{2y^2 - 18y + 40} + \frac{6y - 10}{2y^2 - 18y + 40} = \frac{(14y + 8) + (6y - 10)}{2y^2 - 18y + 40} $$

Step 5: Simplify the resulting expression

Evaluate the combined numerator and simplify the expression: $$ \frac{14y + 8 + 6y - 10}{2y^2 - 18y + 40} = \frac{20y - 2}{2y^2 - 18y + 40} $$ Now, factor out the common factor of 2 from the numerator: $$ \frac{2(10y - 1)}{2y^2 - 18y + 40} $$ Finally, cancel the common factor of 2 from both the numerator and the denominator: $$ \frac{10y - 1}{y^2 - 9y + 20} $$ So the simplified result of the given exercise is: $$ \frac{10y - 1}{y^2 - 9y + 20} $$

Key Concepts

Least Common Denominator

When handling rational expressions, finding the Least Common Denominator (LCD) is crucial. Let's understand why! If you want to add or subtract fractions, they must share the same denominator, known as their common denominator. The **least common denominator** is the smallest shared multiple of the denominators.
In this exercise, we found the LCD between two expressions:

• First denominator: \(6y^2 - 32y + 32\)
• Second denominator: \(2y^2 - 18y + 40\)

Interestingly, the LCD is often not both denominators multiplied together. Instead, it's more efficient, often a whole multiple of one that's easily derived, as seen here. The second denominator was determined as the LCD because it was twice as large as the first, and thus encompassed both expressions efficiently. Finding this LCD allowed us to convert both rational expressions to share this common denominator, simplifying our addition process dramatically.

Numerator and Denominator

To tackle rational expressions, it's vital to be able to identify and work with the numerator and denominator.
The numerator is the top part of the fraction, and the denominator is the bottom. These two components define the fraction and its value. For instance, in our exercise:

• First Expression's numerator: \(7y + 4\)
• First Expression's denominator: \(6y^2 - 32y + 32\)
• Second Expression's numerator: \(6y - 10\)
• Second Expression's denominator: \(2y^2 - 18y + 40\)

Changing these parts affects the fraction value significantly. By multiplying both the numerator and the denominator by the same number (such as \(2\) for the first expression), we create what’s called an "equivalency," ensuring we don't actually change the fraction's value. This is essential when we want to combine expressions, as they need matching denominators to proceed further.

Simplifying Expressions

Simplifying expressions is an essential step that involves reducing expressions to their simplest form. It can make complex rational expressions easier to work with and understand.
In our example problem, we initially combined the numerators after establishing a common denominator. The Combined expression was:

• Numerator: \(14y + 8 + 6y - 10\) simplified to \(20y - 2\)
• Denominator: \(2y^2 - 18y + 40\)

Once combined, our next goal was to simplify this further. We factored out the common factor of \(2\) from the numerator, resulting in \(2(10y-1)\). This made it possible to cancel out the common factor in both the numerator and denominator.
Through this process, we landed on a fully simplified result, making it much cleaner and easier to interpret: \(\frac{10y - 1}{y^2 - 9y + 20}\). Simplifying expressions not only helps to clear the clutter but often uncovers more about the expression's characteristics.