Problem 15 - 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. $$7 x-5 x+6-4$$

Step-by-Step Solution

Answer

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

Step 1: Rearrange Terms Using the Commutative Property

Rewrite the expression by grouping similar terms together. The given expression is: \[ 7x - 5x + 6 - 4 \] Use the commutative property to rearrange terms so that like terms are adjacent: \[ (7x - 5x) + (6 - 4) \]

Step 2: Combine the Like Terms

Now combine the like terms. For the variable terms: Combine \( 7x \) and \( -5x \): \[ 7x - 5x = 2x \] For the constant terms: Combine \( 6 \) and \( -4 \): \[ 6 - 4 = 2 \].

Step 3: Write the Final Simplified Expression

Now write down the result by putting together the simplified variable term and constant: \[ 2x + 2 \] This is your final simplified expression.

Key Concepts

Commutative Property

The commutative property is a fundamental concept in mathematics that allows you to change the order of numbers and variables in an expression without changing its value. This property is useful when simplifying algebraic expressions because it helps to rearrange terms in a way that makes the process of combining like terms easier. Specifically, the commutative property applies to both addition and multiplication operations. For instance, when we rearrange terms like in the expression \(7x - 5x + 6 - 4\), we are effectively using the commutative property. This allows us to group like terms together, which sets the stage for easier simplification. Remember, the order doesn't affect the sum or product:

• For addition: \(a + b = b + a\)
• For multiplication: \(a \times b = b \times a\)

In our case, rearranging the expression by making like terms neighbors - \((7x - 5x) + (6 - 4)\) - is a classic application of this property, paving the way for combining like terms.

Combining Like Terms

Combining like terms is a technique used to simplify algebraic expressions. Like terms are terms in an expression that contain the same variable raised to the same power. These terms can be added or subtracted from each other easily. In our expression \((7x - 5x) + (6 - 4)\), both \(7x\) and \(-5x\) are considered like terms because they both contain the variable \(x\) raised to the first power. Similarly, \(6\) and \(-4\) are like terms as they are both constants (terms with no variable). By combining these like terms, we simplify the expression:

• \(7x - 5x\) simplifies to \(2x\),
• and \(6 - 4\) simplifies to \(2\).

So, the entire expression reduces to \(2x + 2\). This technique not only simplifies the expression but also makes it easier to analyze and solve in further algebraic processes.

Prealgebra Concepts

Prealgebra lays the foundation for all future math learning, and understanding how to simplify expressions is a key part of this foundation. When tackling expressions like the one in the given exercise, the skills and concepts from prealgebra directly apply.Some fundamental prealgebra concepts include:

• Understanding variables and how they can represent unknown numbers.
• Being able to rearrange terms and group like terms using properties like the commutative property.
• Performing arithmetic operations with both integers and rational numbers effectively.

By mastering these skills, students are equipped to simplify complex algebraic expressions into more manageable forms. In our example, we used these concepts to transform \(7x - 5x + 6 - 4\) into the simpler expression \(2x + 2\). Developing a solid grasp of prealgebra not only aids in simplifying expressions but also supports a stronger understanding of more advanced algebraic concepts.