Problem 32 - 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. $$6 a-4-2 a+6 a$$

Step-by-Step Solution

Answer

The simplified expression is \(10a - 4\).

Step 1: Identify Like Terms

Like terms are terms that contain the same variables raised to the same powers. In the expression \(6a - 4 - 2a + 6a\), the like terms are \(6a\), \(-2a\), and \(6a\) because they all have the variable \(a\). The \(-4\) is a constant term with no variable.

Step 2: Rearrange Terms Using Commutative Property

We can use the commutative property to rearrange the terms so that like terms are next to each other. The expression becomes \(6a + 6a - 2a - 4\).

Step 3: Combine Like Terms

Add or subtract the coefficients of the like terms. For the terms with \(a\), combine \(6a + 6a - 2a\): \(6 + 6 - 2 = 10\). The expression simplifies to \(10a - 4\).

Step 4: Final Simplified Expression

The resulting expression after combining all like terms is \(10a - 4\). This is the simplest form of the original expression.

Key Concepts

Understanding Combining Like Terms

When we talk about combining like terms, we're essentially trying to make a complex math expression simpler. Like terms are terms that contain the same variable components with the same exponents. This means that in an expression like \(6a - 4 - 2a + 6a\), the terms \(6a\), \(-2a\), and another \(6a\) are considered like terms. They're like siblings with the same last name (or variable in this case, \(a\)). They can be added or subtracted from one another because they share the same variable trait.

• Like Terms: Terms with identical variable parts.
• Combine Like Terms: Add or subtract their coefficients.
• Constants: Numbers without variables (like \(-4\)) are handled separately.

Recognizing and combining these like terms is crucial because it reduces the complexity of expressions. In our example, the like terms can be simplified together to form \(10a\), making the expression much easier to work with and understand.

Using the Commutative Property

The commutative property is a nice trick in math that lets us rearrange terms in an expression without changing its value. Imagine you're arranging furniture in a room; you can move your couch from one wall to another, but the room is still the same. This property applies similarly to addition and multiplication.

• Switching Order: In expressions like \(ab + ba\), you can reorder to form \(ba + ab\).
• No Change in Value: Reordering doesn't change the overall sum or product.

For example, in the original expression \(6a - 4 - 2a + 6a\), using the commutative property, we can rearrange it to \(6a + 6a - 2a - 4\). This rearrangement makes it clearer to identify and combine like terms next to each other, streamlining the simplification process.

Simplifying Expressions Effortlessly

Simplifying expressions is like cleaning up a messy room; it makes everything more organized and easier to use. By simplifying, we reduce a complex expression into the simplest form possible so that it's straightforward to understand and use in problem-solving.

• Identify Components: Recognize like terms and constants.
• Rearrange: Use properties like the commutative property for easy grouping.
• Combine: Simplify by adding/subtracting coefficients of like terms.

In our example, after combining like terms after rearrangement, we arrive at a much cleaner expression, \(10a - 4\). This is the simplest form of the expression, making calculations and further manipulations with this expression much easier to manage.