Problem 7 - Algebra 2 and Trigonometry

Question

In \(3-20\) , perform the indicated additions or subtractions and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{y+2}{2}+\frac{2 y-3}{3} $$

Step-by-Step Solution

Answer

\( \frac{7y}{6} \) with no undefined values for \( y \).

Step 1: Find the Least Common Denominator (LCD)

To add fractions, we first need to find a common denominator. The denominators here are 2 and 3. The least common denominator of 2 and 3 is 6.

Step 2: Rewrite Each Fraction Using the LCD

Adjust each fraction to have the common denominator of 6. Multiply both the numerator and the denominator of the first fraction by 3: \( \frac{y+2}{2} \times \frac{3}{3} = \frac{3(y+2)}{6} = \frac{3y+6}{6} \). Multiply both the numerator and the denominator of the second fraction by 2: \( \frac{2y-3}{3} \times \frac{2}{2} = \frac{2(2y-3)}{6} = \frac{4y-6}{6} \).

Step 3: Add the Fractions

Now that the fractions have the same denominator, add the numerators: \( \frac{3y+6}{6} + \frac{4y-6}{6} = \frac{3y+6 + 4y-6}{6} = \frac{7y}{6} \).

Step 4: Simplify the Result

The simplified form of the expression is \( \frac{7y}{6} \). This expression cannot be simplified further.

Step 5: Identify Undefined Values

To find when the original fractions are undefined, set each denominator equal to zero and solve. The first fraction’s denominator 2 is never zero, and neither is the second fraction's denominator 3. Therefore, there are no values of \( y \) for which either fraction is undefined.

Key Concepts

Least Common Denominator

When dealing with algebraic fractions, finding the Least Common Denominator (LCD) is crucial. It's the smallest number that both denominators can divide into without leaving a remainder. In this exercise, we are working with the denominators 2 and 3. Finding the LCD involves identifying the smallest multiple that both denominators share.
The multiples of 2 are 2, 4, 6, 8, and so on, while the multiples of 3 are 3, 6, 9, and so on. By comparing these lists, we see that 6 is the smallest multiple they both have in common. It becomes the LCD.
This step is necessary because it allows us to rewrite each fraction so they have a common base, making it simple to perform the addition or subtraction of fractions. Without a common denominator, these operations wouldn't be mathematically consistent or accurate.

Simplification of Fractions

Simplification makes fractions easier to understand and work with. Once we have a common denominator, the next step is to manipulate each fraction so they share this common base. This adjustment involves multiplying both the numerator and the denominator by whatever number it takes to get that common denominator.

• For the fraction \( \frac{y+2}{2} \), we multiply by \( \frac{3}{3} \) to get \( \frac{3(y+2)}{6} \), which simplifies to \( \frac{3y+6}{6} \).
• For the fraction \( \frac{2y-3}{3} \), we multiply by \( \frac{2}{2} \) to achieve \( \frac{4y-6}{6} \).

This manipulation results in fractions with the same denominator that can be easily added together. Once added, if possible, you should further simplify the result.
In our exercise, adding \( \frac{3y+6}{6} \) and \( \frac{4y-6}{6} \) gives us \( \frac{7y}{6} \), which is already in its simplest form.

Undefined Values in Fractions

Fractions are only defined when their denominators are non-zero. In algebra, identifying these undefined values ensures we avoid mathematical errors like division by zero, which is not permissible.
To determine where fractions become undefined, set each denominator equal to zero and solve for the variable. In our case, the denominators are 2 and 3. Since neither is dependent on the variable \( y \), there are no values of \( y \) that result in a zero denominator. Thus, both fractions are defined for all real numbers.
Understanding when fractions become undefined is imperative in mathematics. It avoids incorrect solutions and ensures that mathematical operations are valid across all possible values of the variables involved.