Problem 35 - Elementary Algebra

Question

Simplify. $$ 2 x-3 x $$

Step-by-Step Solution

Answer

-x

Step 1: Combine Like Terms

The expression given is \(2x - 3x\). Both terms are 'like terms' because they are multiples of \(x\). The expression can be combined by subtracting the coefficients of \(x\). This means we perform the operation \(2 - 3\), which results in \(-1\).

Step 2: Write the Simplified Expression

After finding the coefficient by summing \(2 - 3 = -1\), the simplified expression becomes \(-1x\). In standard form, it is usually written as \(-x\). Thus, \(2x - 3x = -x\).

Key Concepts

Like Terms

In algebra, 'like terms' are terms that contain the same variable raised to the same power. Identifying like terms is crucial for simplifying expressions, as it allows us to combine them effectively. For instance, consider the expression \(2x - 3x\). Here, both terms contain the variable \(x\) with the same power, which makes them like terms. Thus, they can be joined together by operating on their coefficients.

• Like terms must have identical variables and exponents.
• Example: \(5y\) and \(-2y\) are like terms, but \(5y^2\) and \(5y\) are not.
• Identifying like terms makes algebraic simplification easier.

Recognizing like terms is the first step to efficiently simplifying algebraic expressions.

Coefficients

The concept of 'coefficients' is foundational in algebra. A coefficient is a number that multiplies a variable in an algebraic expression. In the expression \(2x - 3x\), the numbers 2 and -3 are coefficients. They indicate how many times the variable \(x\) is taken into account.

• A coefficient is often an integer but can also be a fraction or decimal.
• The expression \(a \cdot b\) has 'a' as the coefficient of 'b'.

When combining like terms, we focus on adding or subtracting the coefficients. This step leads us towards simplifying the whole expression. In our example, performing \(2 - 3\) gives us the combined coefficient of \(-1\) for \(x\). Thus, understanding coefficients allows us to restructure and simplify expressions effectively.

Algebraic Simplification

Algebraic simplification is the process of reducing an expression to its simplest form. In this context, simplifying involves combining like terms and optimizing the way terms in an expression are represented. For \(2x - 3x\), simplifying means performing the math on the coefficients to combine the terms effectively.

• Start by identifying like terms within the expression.
• Operate on the coefficients of the like terms.
• Rewrite the expression using the newly found coefficients.

In our example, subtracting the coefficients, \(2 - 3 = -1\), gives us the simplified term \(-x\). Doing so helps in making the expression more manageable and understandable.By comprehensively simplifying algebraic expressions, you create versions that are easier to interpret and utilize in predictions or problem-solving.