Problem 6 - 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. $$ \frac{a-1}{5}-\frac{a+1}{4} $$

Step-by-Step Solution

Answer

The result in simplest form is \( \frac{-a - 9}{20} \). The expression is defined for all real numbers of \(a\).

Step 1: Find Common Denominator

To subtract fractions, find a common denominator. The denominators are 5 and 4. The least common multiple of 5 and 4 is 20. So, the common denominator is 20.

Step 2: Rewrite Fractions with Common Denominator

Rewrite each fraction with 20 as the denominator. For \( \frac{a-1}{5} \), multiply both the numerator and denominator by 4 to get \( \frac{4(a-1)}{20} \). For \( \frac{a+1}{4} \), multiply both the numerator and denominator by 5 to get \( \frac{5(a+1)}{20} \).

Step 3: Perform Subtraction

Subtract the second fraction from the first, now that they have a common denominator: \( \frac{4(a-1)}{20} - \frac{5(a+1)}{20} = \frac{4(a-1) - 5(a+1)}{20} \).

Step 4: Simplify the Numerator

Expand and simplify the numerator: \( 4(a-1) - 5(a+1) = 4a - 4 - 5a - 5 = -a - 9 \).

Step 5: Write the Result in Simplest Form

Now combine the terms to write the expression as \( \frac{-a - 9}{20} \). This is in simplest form since there are no common factors between the numerator and the numerator's terms.

Step 6: Identify Undefined Values

Fractions are undefined when the denominator is zero. However, in this fraction, the denominator is a constant (20), meaning it is never zero, and there are no restrictions on the variable values.

Key Concepts

Common Denominator

When performing arithmetic operations, such as addition or subtraction on fractions, it's essential to have a common denominator. A common denominator is a shared multiple of the denominators in the fractions you are working with. Considering our problem, the denominators are 5 and 4. To find a common denominator, we look for the Least Common Multiple (LCM) of these numbers. In this case, the LCM of 5 and 4 is 20. By representing both fractions with this common denominator, we can combine them easily. Without a common denominator, direct addition or subtraction is not possible.

Subtraction of Fractions

Subtracting fractions involves a simple but structured process. Once we identify a common denominator, we rewrite each fraction to reflect this. For the subtraction \[ \frac{a-1}{5} - \frac{a+1}{4} \]we change each fraction to have a denominator of 20:- Multiply the numerator and denominator of \( \frac{a-1}{5} \) by 4: \( \frac{4(a-1)}{20} \)- Multiply the numerator and denominator of \( \frac{a+1}{4} \) by 5: \( \frac{5(a+1)}{20} \)Now, both fractions have the same denominator of 20, allowing us to subtract their numerators directly:\[ \frac{4(a-1)}{20} - \frac{5(a+1)}{20} = \frac{4(a-1) - 5(a+1)}{20} \]. The key is maintaining the common denominator while calculating the difference in numerators.

Simplifying Expressions

Simplifying expressions is a process of combining and reducing terms to their simplest form. After performing subtraction:\[ \frac{4(a-1) - 5(a+1)}{20} \] it's time to simplify the numerator. Begin by distributing the numbers:- Expand \(4(a - 1)\) to get \(4a - 4\)- Expand \(5(a + 1)\) to get \(5a + 5\)Next, combine like terms:\[ 4a - 4 - 5a - 5 = -a - 9 \]Thus, the expression simplifies to:\[ \frac{-a - 9}{20} \]The numerator is neatly simplified, and the fraction itself is in its simplest form as no further reduction is possible.

Undefined Values in Fractions

Fractions can become undefined if their denominators are zero because division by zero is not permissible in mathematics. To determine if a fraction has undefined values, we check the denominator.In our problem, the final expression is:\[ \frac{-a - 9}{20} \]Here, the denominator is a constant value (20), and it's never zero. Therefore, there are no restrictions on the variable \(a\); it can be any real number. If the denominator were a variable expression, we would solve for when it equals zero to find the undefined values.
Being aware of undefined values helps in understanding the domain of variable expressions.