Problem 92 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. $$ -\frac{5}{18} x-\frac{7}{18} x $$

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{2}{3}x\).

Step 1: Identify Like Terms

First, recognize that both terms in the expression \[-\frac{5}{18}x - \frac{7}{18}x\] are like terms because they both have the same variable portion, which is \(x\).

Step 2: Combine Coefficients

Now, add the coefficients of the like terms together. The coefficients are \(-\frac{5}{18}\) and \(-\frac{7}{18}\). Therefore, we calculate: \[-\frac{5}{18} + -\frac{7}{18} = -\left(\frac{5}{18} + \frac{7}{18}\right) = -\frac{12}{18}\].

Step 3: Simplify the Fraction

The fraction \(-\frac{12}{18}\) can be simplified. Determine the greatest common divisor, which is 6, and divide both the numerator and the denominator by it: \[-\frac{12}{18} = -\frac{12 \div 6}{18 \div 6} = -\frac{2}{3}\].

Step 4: Rewrite the Expression

Rewrite the original expression using the simplified coefficient: \[-\frac{2}{3}x\].This is the simplified version of the expression.

Key Concepts

Like Terms

When working with algebraic expressions, identifying like terms is a crucial step. In simple terms, like terms are terms that have the same variable components. They can look different at first glance due to different coefficients, but what connects them is their variables. For example, in \[-\frac{5}{18}x - \frac{7}{18}x\], both terms feature the variable \(x\). This makes them like terms. Recognizing like terms is the first step toward simplifying expressions because it allows you to combine them easily.

• Like terms must have the exact same variable raised to the same power.
• Constant terms (terms without variables) are considered like terms with other constant terms.

By properly identifying like terms, you are setting a strong foundation for the next steps in simplifying algebraic expressions. This concept plays an important role in making complex equations more manageable.

Combining Coefficients

Once you have identified like terms in an expression, the next step is to combine their coefficients. The coefficients are the numerical parts of the terms. In our example \[-\frac{5}{18}x - \frac{7}{18}x\], \(-\frac{5}{18}\) and \(-\frac{7}{18}\) are the coefficients. To combine them, you simply add them together, keeping in mind the rules for adding negative numbers.

• First, remove the variable temporarily and focus solely on the coefficients.
• Add the coefficients: \(-\frac{5}{18} + -\frac{7}{18}\).
• This simplifies to \(-\frac{12}{18}\).

Combining coefficients is an essential skill because it directly reduces clutter in your expression, making it easier to read and understand. Always pay attention to the signs (positive or negative) when combining coefficients, as they affect the outcome.

Simplifying Fractions

Simplifying fractions is an important step in achieving the most reduced form of an expression. After you've combined coefficients in the expression, you may end up with a fraction that can be simplified further. Take our result: \(-\frac{12}{18}\).To simplify a fraction, divide both the numerator and denominator by their greatest common divisor (GCD).

• Find the GCD of 12 and 18, which is 6.
• Divide both numerator and denominator by 6: \(-\frac{12}{6} = -2\) and \(\frac{18}{6} = 3\).
• This gives you the simplified fraction: \(-\frac{2}{3}\).

Knowing how to simplify fractions ensures your final answer is neat and easy to understand. It also helps in further operations you might need to do with the expression. Simplified fractions are often easier to work with in equations, making them a valuable aspect of algebraic manipulation.