Problem 29 - 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. $$12 y+3+5 y$$

Step-by-Step Solution

Answer

The simplified expression is \(17y + 3\).

Step 1: Identify Like Terms

The given expression is \(12y + 3 + 5y\). First, identify the like terms in the expression. Like terms are terms that have the same variable raised to the same power. Here, like terms are \(12y\) and \(5y\), because they both contain the variable \(y\). The \(3\) is a constant term and stands alone.

Step 2: Use the Commutative Property to Rearrange Terms

Rearrange the terms using the commutative property of addition, which says you can change the order of terms when adding them. Rearrange the expression \(12y + 3 + 5y\) as \(12y + 5y + 3\) to place the like terms next to each other.

Step 3: Combine Like Terms

Now, combine the like terms. Add \(12y\) and \(5y\) together. The coefficients of the \(y\)-terms are \(12\) and \(5\), so add them: \(12 + 5 = 17\). Therefore, \(12y + 5y = 17y\). The constant \(3\) remains unchanged.

Step 4: Write the Simplified Expression

The simplified expression is formed by the combined like terms and the constant. After combining, the expression \(12y + 3 + 5y\) becomes \(17y + 3\). Thus this is your simplified expression.

Key Concepts

Like Terms

When you're dealing with algebraic expressions, **like terms** are a fundamental concept that can make simplifying expressions much easier. But what exactly are like terms? Simply put, they are terms in an expression that have the same variables raised to the same powers. For instance, in the expression \( 12y + 3 + 5y \), both \( 12y \) and \( 5y \) are like terms because they contain the variable \( y \) raised to the first power. The constant \( 3 \), on the other hand, is a standalone term because it has no variable attached to it.To identify like terms, look for:

• The same variable in each term.
• Variables raised to the same power.

Understanding and identifying like terms is crucial because once identified, you can simplify the expression by combining them, making it simpler and easier to work with.

Commutative Property

The **commutative property** is a handy tool in mathematics that applies to both addition and multiplication. This property tells us that when performing operations like addition or multiplication, the order of the operands doesn't change the result. For the expression \( 12y + 3 + 5y \), this property allows us to rearrange terms so that it's easier to combine like terms.For example, according to the commutative property of addition, you can rearrange the terms as follows:

• Original order: \( 12y + 3 + 5y \)
• Rearranged order using commutative property: \( 12y + 5y + 3 \)

This flexibility is particularly useful in simplifying expressions because it allows us to place like terms next to each other, making the process of combining them straightforward and more intuitive.

Combining Like Terms

Once you’ve identified and appropriately arranged the like terms in your expression, you can perform the step of **combining like terms**. This is the step where the actual simplification happens, transforming an expression into its simplest form.Continuing with the expression \( 12y + 5y + 3 \), we see that the like terms, \( 12y \) and \( 5y \), share the variable \( y \). To combine them:

• Add the coefficients (the numbers in front of the variables): \( 12 + 5 = 17 \).
• The expression with combined like terms becomes \( 17y \).

The constant \( 3 \) remains unchanged because it doesn’t have any like terms to combine with. So, after combining, the simplified expression is \( 17y + 3 \). This process not only reduces the number of terms in an expression but also makes it clearer and easier to understand.