Problem 19 - 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. $$6 y-2 y-5+1$$

Step-by-Step Solution

Answer

The simplified expression is \(4y - 4\).

Step 1: Identify Like Terms

To simplify the expression, identify the similar terms. In the expression \(6y - 2y - 5 + 1\), the like terms are those with the variable \(y\) and the constants. Thus, \(6y\) and \(-2y\) are like terms, as well as \(-5\) and \(+1\).

Step 2: Rearrange Using the Commutative Property

Reorder the expression to group like terms together. Using the commutative property, rearrange the expression to look like this: \(6y - 2y + (-5 + 1)\).

Step 3: Combine the Like Terms

Now, add the coefficients of the \(y\) terms together: \(6y - 2y = 4y\). Then combine the constant terms: \(-5 + 1 = -4\).

Step 4: Write the Simplified Expression

Write down the simplified expression using the results from the previous step. Thus, the expression \(6y - 2y - 5 + 1\) simplifies to \(4y - 4\).

Key Concepts

Understanding Like Terms

When we mention 'like terms,' we're talking about components of a mathematical expression that have the same variables raised to the same powers. In other words, they are items that can be directly combined. For instance, in the expression \(6y - 2y - 5 + 1\), both \(6y\) and \(-2y\) are like terms because they have the same variable \(y\). Similarly, \(-5\) and \(+1\) are also like terms because they are both constant numbers without any variables.
Identifying like terms is the first step in the simplification process because it allows you to combine them, which makes your expression simpler and more concise. Always remember that terms can only be combined if they are 'like':

• Same variable with the same exponent.
• Two or more constant terms.

By isolating like terms through careful identification, you set the stage for simplifying expressions effectively.

Exploring the Commutative Property

The commutative property is a key principle in algebra that refers to the ability to swap numbers around in an addition or multiplication equation without changing the result. This property is very useful when simplifying expressions, as it lets you rearrange terms.
In the original expression \(6y - 2y - 5 + 1\), using the commutative property means you can switch around the terms so that like terms are next to each other, making them easier to combine. So, you can rearrange it as \(6y - 2y + (-5 + 1)\). This step sets up the equation nicely for the next phase of simplification.

• The commutative property applies to addition: \(a + b = b + a\).
• It also applies to multiplication: \(a \cdot b = b \cdot a\).

Understanding and effectively applying the commutative property simplifies expression simplification tasks.

Combining Terms for Simplification

Once like terms have been identified and the commutative property has been used to arrange them, the next step is straightforward: combining the terms. This significantly reduces the complexity of the expression and gets us closer to the final simplified form.
For our example expression, \(6y - 2y + (-5 + 1)\), we start by adding the coefficients of the like terms. The terms \(6y\) and \(-2y\) are combined by subtracting: \(6y - 2y = 4y\). After that, you combine the constants \(-5\) and \(+1\) by adding: \(-5 + 1 = -4\). The simplified expression is thus \(4y - 4\).

• Always combine coefficients of like terms first.
• Ensure you've grouped all like terms before adding or subtracting.
• Recheck your work to confirm no terms are overlooked.

Combining like terms is a critical part of simplification, resulting in a cleaner and undeviating expression.