Problem 62 - Beginning and Intermediate Algebra

Question

Simplify each expression. $$ w+(-w)+\frac{1}{4}(4) $$

Step-by-Step Solution

Answer

1

Step 1: Combine Like Terms

Look at the expression: \(w + (-w) + \frac{1}{4}(4)\). Notice that there is \(w\) and \(-w\) in the expression. Combining these terms yields 0 since \(w - w = 0\). This simplifies the expression to \(0 + \frac{1}{4}(4)\).

Step 2: Simplify the Fraction

Now, simplify \(\frac{1}{4} \cdot 4\). Multiply \(\frac{1}{4}\) by 4: \(\frac{1}{4} \cdot 4 = 1\). This gives us: \(0 + 1\).

Step 3: Combine Final Terms

Combine 0 and 1: \(0 + 1 = 1\). Therefore, the simplified expression is 1.

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, it's crucial to combine like terms. Like terms are terms that have the same variable raised to the same power. For instance, in the expression \(w + (-w) + \frac{1}{4}(4)\), the terms \(w\) and \(-w\) are like terms because they both contain the variable \(w\).
Combining these terms simplifies the expression:

• \(w - w = 0\)

Now the expression becomes:
\(0 + \frac{1}{4}(4)\).
Combining like terms helps in reducing complex expressions to simpler forms, making them easier to work with.

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its simplest form where the numerator and the denominator are as small as possible. This makes the fraction easier to understand and work with. In our example, the term \(\frac{1}{4}(4)\) needs to be simplified.
Multiply the fraction by the number:

• \(\frac{1}{4} \cdot 4 = 1\)

Hence, the simplified term is 1.
Simplifying fractions is essential in algebra as it streamlines calculations and helps in better comprehension of the expressions.

Algebra Basics

Understanding the basics of algebra is essential in solving and simplifying expressions. Algebra involves using symbols and letters to represent numbers and quantities in formulas and equations.
In this exercise, we used foundational concepts:

• Combining like terms: Reducing terms with the same variable.
• Simplifying fractions: Making fractions as simple as possible.

After combining like terms and simplifying fractions, our final step is to combine the remaining simplified terms: \(0 + 1 = 1\).
Therefore, the final simplified expression is 1. Mastering these basics ensures a strong foundation for tackling more complex algebraic problems.
Remember to always break down the problem step by step, which makes solving easier and less error-prone.