Problem 91 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. $$ -\frac{7}{16} x-\frac{3}{16} x $$

Step-by-Step Solution

Answer

The simplified expression is \\(-\frac{5}{8}x\\).

Step 1: Identify Like Terms

Both terms in the expression \(-\frac{7}{16}x - \frac{3}{16}x\) have the variable \(x\) and are over a common denominator (16). Hence, they are like terms and can be combined by addition or subtraction.

Step 2: Combine Like Terms

Since the denominators are the same, combine the numerators by performing the operation indicated: \(-\frac{7}{16} - \frac{3}{16} = -\frac{(7+3)}{16} = -\frac{10}{16}\).

Step 3: Simplify the Fraction

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2. Thus, \(-\frac{10}{16} = -\frac{5}{8}\).

Step 4: Finalize the Expression

The expression simplifies to \(-\frac{5}{8}x\). This is the simplest form of the given expression.

Key Concepts

Like Terms

Like terms are terms in an algebraic expression that have the same variables raised to the same power. In our example, \(-\frac{7}{16}x\) and \(-\frac{3}{16}x\) are like terms because they both contain the variable \(x\). This similarity allows us to combine them easily when simplifying expressions.
When dealing with equations or simplifying expressions, always look for like terms first. They make the initial steps of solving or simplifying much more straightforward.

• Identify terms with the same variable(s) and power(s).
• Combine them using addition or subtraction.

Combining like terms reduces complexity and gives a cleaner, more understandable expression.

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. In our problem, both terms \(-\frac{7}{16}x\) and \(-\frac{3}{16}x\) have the same denominator, 16. This makes adding or subtracting the fractions much easier.
When fractions share a common denominator:

• You can directly perform addition or subtraction on their numerators, keeping the denominator unchanged.
• This is essential for efficiently combining fractions when simplifying expressions.

Always check if fractions have a common denominator before performing operations, as it simplifies the process significantly.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. This happens by dividing both the numerator and the denominator by their greatest common divisor (GCD). In our example, we simplified \(-\frac{10}{16}\) by dividing both 10 and 16 by their GCD, which is 2, resulting in \(-\frac{5}{8}\).
Steps to simplify a fraction:

• Find the GCD of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.
• The result is the fraction in its simplest form.

Simplifying fractions is crucial because it reduces fractions to their most essential representation, making them much easier to understand and work with.