Problem 27 - Prealgebra

Question

Simplify the following expressions by combining similar terms. In some cases the order of the terms must be rearranged first by using the commutative property. $$3 x+2-4 x+1$$

Step-by-Step Solution

Answer

The expression simplifies to \(-x + 3\).

Step 1: Identify Like Terms

Identify the terms in the expression that are similar. Like terms have the same variable part. For the expression \(3x + 2 - 4x + 1\), the like terms are \(3x\) and \(-4x\), as they both contain the variable \(x\). Similarly, \(2\) and \(1\) are like terms as they are both constants.

Step 2: Rearrange Using the Commutative Property

Rearrange the expression by grouping the like terms together. Using the commutative property, where the order of addition can be changed, the expression \(3x + 2 - 4x + 1\) can be rewritten as \(3x - 4x + 2 + 1\).

Step 3: Combine the Like Terms with Variables

Add or subtract the coefficients of the like terms with the variable \(x\). Combine \(3x\) and \(-4x\): \(3x - 4x = -1x\). Thus, the expression becomes \(-x + 2 + 1\).

Step 4: Combine the Constant Terms

Add the constant terms together. Take the remaining constants \(2\) and \(1\) and combine them: \(2 + 1 = 3\). The expression is now \(-x + 3\).

Key Concepts

Understanding the Commutative Property

When simplifying algebraic expressions, the commutative property of addition can be a powerful tool. This mathematical rule states that numbers can be added in any order without changing the sum. If we look at our expression, which is initially written as \(3x + 2 - 4x + 1\), we can rearrange the like terms. For instance, shifting the order to group similar terms together means rewriting it as \(3x - 4x + 2 + 1\). By restructuring the expression this way, calculations become easier. The commutative property is especially useful in more complex expressions, allowing us to rearrange terms to simplify our arithmetic.

• Remember: The order of addition does not affect the result.
• Use it to make complex problems more manageable.
• It's like organizing your thoughts for simpler processing.

Variable Terms and Their Role

Variable terms are parts of an expression containing a variable, such as \(x\). In our given expression, \(3x\) and \(-4x\) are the variable terms because they both include the variable \(x\). Identifying these terms is crucial because they will be combined based on their coefficients when simplifying. The expression \(3x - 4x\) simplifies to \(-x\) because we perform the operation \(3 - 4\) on the coefficients.

• Variable terms include letters like \(x\), \(y\), etc.
• They represent unknown values that can change.
• Combine them by adding/subtracting their coefficients.

Understanding variable terms allows you to manage the unknowns in algebra effectively.
Each operation is about dealing with those coefficients first.

Incorporating Constants Into the Mix

Constants are terms in an algebraic expression that do not contain variables. They are the stable numbers in the equation. In our example, the constants are \(2\) and \(1\). Once variable terms are combined, constants can be simplified by performing straightforward arithmetic. For example, combining \(2 + 1\) results in the constant \(3\). This is key to reaching the simplified form of any expression.

• Constants remain fixed and unchanging.
• Simplify them by regular addition or subtraction.
• They help ground the expression in known values.

By fully simplifying each part, you finish with an expression like \(-x + 3\). Constants are essential for finding the final numeric part of an expression.