Problem 98 - Intermediate Algebra

Question

Simplify each expression. $$-\frac{5}{9} y-\frac{7}{18} y$$

Step-by-Step Solution

Answer

The simplified expression is \\(-\frac{17}{18}y\\).

Step 1: Find a Common Denominator

The first step is to find a common denominator for the fractions. The denominators are 9 and 18, and the least common denominator (LCD) is 18.

Step 2: Convert Fractions to the Common Denominator

Convert \(-\frac{5}{9}\) to the common denominator of 18. Multiply the numerator and the denominator by 2: \(-\frac{5}{9} = -\frac{10}{18}\).

Step 3: Combine the Like Terms

Now combine the fractions by adding their numerators since they have the same denominator: \(-\frac{10}{18}y - \frac{7}{18}y = -\frac{17}{18}y\).

Step 4: Simplified Expression

The expression simplified is \(-\frac{17}{18}y\).

Key Concepts

Common Denominator

When working with fractions, especially when adding or subtracting them, it's important to have a common denominator. This means that both fractions should have the same bottom number. By doing this, it becomes much easier to perform the operation.

In the expression \(-\frac{5}{9} y - \frac{7}{18} y\), the denominators are 9 and 18. To find a common denominator, follow these steps:

• Identify the denominators in the fractions.
• Find the smallest number that both denominators can divide into evenly. This is called the Least Common Denominator (LCD).

In our example, 18 is the smallest number that both 9 and 18 can divide into without leaving a remainder, making it the common denominator. Once both fractions share the same denominator, we can proceed with the rest of the calculation.

Combining Like Terms

When an expression has terms with the same variable raised to the same power, these terms are called "like terms." In our example, \(-\frac{5}{9} y\) and \(-\frac{7}{18} y\), the terms are like terms because both contain the variable \(y\).

After converting the fractions to have a common denominator, you can easily combine them. Here's how:

• Ensure both terms share the same denominator as we've previously established.
• Add or subtract the coefficients (numerical part of the terms) and keep the variable part the same.

For instance, in the expression \(-\frac{10}{18}y - \frac{7}{18}y\), you'll add the coefficients \(-10\) and \(-7\) to get \(-17\). The simplified expression for like terms is \(-\frac{17}{18}y\). This process of combining like terms helps in further simplifying expressions and solving equations.

LCD in Fractions

Finding the least common denominator (LCD) is a crucial step when dealing with expressions that include fractions with different denominators. It makes the process of adding, subtracting, and comparing fractions simpler and more efficient.

Here’s a simple guide on how to find the LCD:

• List the multiples of each denominator.
• Find the smallest multiple that is common to each list.
• This smallest multiple is your LCD.

For the problem \(-\frac{5}{9} y - \frac{7}{18} y\), the denominators are 9 and 18. By listing the multiples, we see that 18 is common to both lists as the smallest number. Only after establishing the LCD, can we convert the fractions to equivalent fractions with this common denominator, thereby allowing us to combine them effectively.