Problem 2 - Algebra for College Students

Question

Simplify the algebraic expressions in Problems \(1-14\) by combining similar terms. $$5 x-8 x+x$$

Step-by-Step Solution

Answer

The simplified expression is \(-2x\).

Step 1: Identify Like Terms

In the expression \(5x - 8x + x\), identify the like terms. Like terms have the same variable raised to the same power. Here, all terms are like terms because they all involve the variable \(x\).

Step 2: Combine Like Terms

Combine the coefficients of the like terms. Here, it means you add the numbers in front of \(x\). Calculate \(5 - 8 + 1\):1. Start with \(5x - 8x\) which is \(-3x\).2. Now, add \(x\) (which is \(+1x\)), so \(-3x + 1x = -2x\).

Step 3: Write the Simplified Expression

After combining the like terms, write down the simplified expression, which is \(-2x\).

Key Concepts

Like Terms

In algebra, like terms are terms that contain the same variables raised to the same power. The coefficients, which are the numbers in front of these variables, can be different. However, the key to identifying like terms is ensuring the variables and their exponents match.
For example, in the expression \(5x - 8x + x\), all terms are like terms because they all contain the variable \(x\) raised to the first power.

• Variable Match: All terms have the variable \(x\).
• Exponents Match: Each \(x\) is raised to the power of 1.

Identifying like terms correctly is the first step towards simplifying algebraic expressions. It's necessary because only like terms can be combined mathematically.

Combining Like Terms

Once you have identified like terms, the next step is to combine them. This involves adding or subtracting the coefficients of the like terms while keeping the variable part unchanged.
For instance, in \(5x - 8x + x\):

• The coefficients are 5, -8, and 1.
• Add or subtract these coefficients as required: compute \(5 - 8 + 1\). This calculation starts with 5 minus 8, giving \(-3\). Then, add 1, resulting in \(-2\).

Thus, the combined expression is \(-2x\). Here, combining like terms helps streamline the expression, making it simpler and easier to work with in further calculations. Using only the matching variable part helps maintain the consistency of the expression while altering only the numerical weights.

Simplification in Algebra

Simplification in algebra is about reducing expressions to their simplest forms while retaining their original meaning and value. This process makes expressions easier to understand and evaluate, especially when solving equations or performing operations with them.
After combining like terms, such as with \(5x - 8x + x\) to get \(-2x\), we arrived at a simplified expression.

• A simplified expression reduces confusion and provides clarity.
• It's also the basis for deeper algebraic understanding—such as solving equations or checking equivalencies between expressions.

In practice, simplification often involves several steps, but a core part is always about unifying like terms first. Thus, while simplification can sometimes seem complex, breaking down into manageable steps like combining coefficients can make it much more approachable.