Problem 28 - Elementary Algebra

Question

Simplify. $$ 1 x-1-2 x $$

Step-by-Step Solution

Answer

-x - 1

Step 1: Identify Like Terms

In the expression, identify the like terms. The terms with variable 'x' are '1x' and '-2x'. The constant term is '-1'.

Step 2: Combine the Like Terms

Now, combine the like terms: The like terms are '1x' and '-2x'. Combine these by summing their coefficients: \[ 1 - 2 = -1 \]Thus, the result for the x terms is '-1x', which simplifies to '-x'.

Step 3: Write the Simplified Expression

After combining the like terms, we have '-x'. The constant term '-1' stays the same. Therefore, the simplified expression is: \[-x - 1\]

Key Concepts

Like Terms

In algebra, **like terms** are terms that have identical variable parts. They can be combined because they have the same variables raised to the same powers. For instance, terms like \(3x\), \(-5x\), and \(x\) are like terms because each term contains the variable \(x\) raised to the first power.
It is important to recognize and categorize like terms in an expression to simplify it by combining them. Unlike terms, such as \(2x\) and \(3y\), cannot be combined as they have different variables.

• Like terms must have the same variable and power.
• Only the coefficients of like terms can be combined.
• Recognizing like terms is essential for algebraic simplification.

Combining Like Terms

The process of **combining like terms** involves adding or subtracting the coefficients of the like terms while keeping the variable part unchanged. This helps in simplifying expressions and making them more manageable.
In the expression \(1x - 2x - 1\), the terms \(1x\) and \(-2x\) are like terms, which means their coefficients can be combined. The coefficients \(1\) and \(-2\) are combined by subtraction to give \(-1\). Therefore, the like terms \(1x\) and \(-2x\) simplify to \(-1x\), commonly written as \(-x\).
Here's how you can combine like terms effectively:

• Identify the like terms in the expression.
• Add or subtract the coefficients of these terms.
• Replace the original like terms with the simplified term.

Applying these steps ensures that the expression is correctly simplified while maintaining its original value.

Algebraic Simplification

**Algebraic simplification** is the process of transforming a complex or lengthy expression into a simpler form. This is done by performing various operations such as combining like terms, factoring, or using algebraic identities.
Simplifying expressions helps to reveal the underlying structure and often makes calculations easier.
In our example, we simplified the expression \(1x - 1 - 2x\) by identifying like terms and combining them to form \(-x\), resulting in a simpler expression \(-x - 1\). This simplified expression is easier to understand and use in further calculations or problem-solving.

• Identify opportunities to combine like terms.
• Apply arithmetic operations on coefficients effectively.
• Aim to reduce the expression to its simplest form for better clarity and usability.

Simplification does not alter the value of the expression but makes it more straightforward to work with, which is a crucial aspect of solving algebra problems efficiently.