Problem 20 - Algebra and Trigonometry

Question

Find the sum, difference, or product. \((5-3 x)+(2 x-8)\)

Step-by-Step Solution

Answer

The simplified expression is \(-3 - x\).

Step 1: Identify Like Terms

Examine the expression \((5 - 3x) + (2x - 8)\) to identify like terms. Like terms are terms that contain the same variable raised to the same power. Here, \(-3x\) and \(2x\) are like terms, and \(5\) and \(-8\) are constant terms.

Step 2: Combine Like Terms

Add the like terms separately. First, combine the constant terms: \(5 - 8 = -3\). Next, combine the terms with \(x\): \(-3x + 2x = -1x\).

Step 3: Write the Simplified Expression

After combining the like terms, rewrite the expression: \(5 - 3x + 2x - 8 = -3 - x\). This is the simplified result of the sum of the given expressions.

Key Concepts

Like Terms

In algebra, one of the key concepts you will encounter is the idea of 'like terms'. Understanding like terms is essential when simplifying algebraic expressions. **What are Like Terms?**Like terms are terms in an algebraic expression that have the exact same variable components. This means they must contain the same variables raised to the same power. For example, in the expression \(3x - 5 + 2x + 7\), the terms \(3x\) and \(2x\) are deemed like terms because they both contain the variable \(x\) raised to the first power. Similarly, the numbers \(-5\) and \(+7\) are like terms since they are both constants.Recognizing like terms enables you to manipulate algebraic expressions more efficiently by combining them, thereby simplifying the expression. This foundational idea will help you in higher-level algebraic operations down the line.

Combine Like Terms

Once you have identified like terms in an algebraic expression, the next step is to 'combine like terms'. This process reduces the expression to its simplest form and makes calculations more straightforward.**How to Combine Like Terms?**To combine like terms, simply add or subtract the coefficients (the numerical parts) of the terms while keeping the variable part the same:

• For example, for \(-3x\) and \(2x\), you combine them as \(-3 + 2\), which gives you \(-1x\) or simply \(-x\).
• With constant terms like \(5\) and \(-8\), you perform the operation \(5 - 8\) to get \(-3\).

Errors can occur if the variables or their powers are not identical; make sure terms are truly alike before combining. This step is crucial in crafting a simplified expression, which is an essential part of solving algebraic problems.

Simplified Expression

After identifying and combining like terms, the final product is known as a 'simplified expression'. This means the expression is as concise as it can possibly be. Simplified expressions are easier to work with in both calculations and further algebraic operations.**Why Simplify?**Simplifying an expression helps in several ways:

• It reduces complexity, making equations easier to solve.
• Simplified expressions cut down on potential errors in calculations.
• They provide a clearer understanding of the underlying relationships between the variables.

In the context of our original problem, the simplified expression from \((5 - 3x) + (2x - 8)\) was \(-3 - x\). This result shows you exactly how the different parts of the original expression interact, without unnecessary clutter. Simplification is a critical skill not just in math but in logical thinking and problem-solving.