Problem 19 - Technical Mathematics with Calculus

Question

Combine and simplify. Try some by calculator. $$\frac{5 x}{2}-\frac{3 x}{2}$$

Step-by-Step Solution

Answer

\(x\)

Step 1: Identify like terms

Determine that both terms have the same denominator and the same variable part, which makes them like terms and suitable for combination.

Step 2: Subtract the numerators

Since the denominators are the same, subtract the numerators while keeping the denominator constant. Perform the operation \(\frac{5x}{2} - \frac{3x}{2} = \frac{5x - 3x}{2}\).

Step 3: Simplify the expression

Combine the numerators which gives \(\frac{2x}{2}\). You can simplify it further by dividing the numerator by the denominator, resulting in \(x\).

Key Concepts

Combining Like Terms

Understanding how to combine like terms is essential in simplifying algebraic expressions. Like terms are terms that have the exact same variable component raised to the same power. For example, in the expression \(5x/2 - 3x/2\), both terms have the variable 'x' to the power of one. To combine them, you simply add or subtract their coefficients (the numerical part in front of the variable), depending on the operation.

When you identify like terms, you consolidate them to make the expression neater and more manageable. This often makes further calculations easier. For example, \(5x/2 - 3x/2\) becomes \(2x/2\), where '5' and '-3' are the coefficients combined through subtraction, resulting in '2'. This concept is not only helpful in simplifying fractions but in all areas of algebra where you work with variables.

Subtracting Fractions

In algebra, subtracting fractions often involves variables as part of the numerators or denominators. To subtract fractions correctly, the key is to ensure they have a common denominator. When working with algebraic fractions like \(5x/2\) and \(3x/2\), since the denominators are the same, you only need to subtract the numerators. This process is also identical to subtracting numerical fractions.

Steps for Subtracting Fractions

• Ensure the fractions have a common denominator.
• Subtract the numerators while the denominators stay the same.
• Simplify the resulting fraction by dividing the numerator by the denominator, if possible.

After subtracting the numerators, the fraction may be simplified further to reduce it to the simplest form, which is a crucial step for clear and concise final answers in algebra.

Algebraic Expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and operation signs. They are the fundamental components of algebra and help to represent relationships between different quantities. Expressions like \(5x/2\) and \(3x/2\) may appear complex but can be simplified by applying the right algebraic operations.

The ability to manipulate these expressions is vital, as it allows for solving equations, modeling situations, and understanding patterns. When you simplify an algebraic expression, you're essentially breaking it down to its most basic form without changing its value, which is exactly what happens when you simplify \(2x/2\) to 'x'. The simplicity of the final expression, 'x', reveals the true power of algebra to express complex relationships in a more understandable way.