Problem 52 - Prealgebra

Question

Use the distributive property to combine similar terms. \(3 y-y\)

Step-by-Step Solution

Answer

The simplified expression is \(2y\).

Step 1: Identify the Expression

The expression we need to simplify is \(3y - y\). Our goal is to combine the similar terms using the distributive property.

Step 2: Use the Distributive Property

Recall the distributive property which states that \(a(b+c) = ab + ac\). Though the terms are already distributed, we can think of \(3y\) as \((3-1)y\) to factor them together.

Step 3: Combine Like Terms

Since both terms have a common variable \(y\), you can factor out the \(y\). This gives \((3-1)y = 2y\). This simplifies the expression to \(2y\).

Key Concepts

Combine Like Terms

When simplifying algebraic expressions, one of the key skills is learning how to combine like terms. This often involves grouping terms that have the same variable raised to the same power, so they can be simplified together.

In our expression, we have two like terms: \(3y\) and \(-y\). Both of these terms have the variable \(y\). A like term is essentially the same variable part, which allows us to combine them by simply adding or subtracting the coefficients.

• The coefficient of \(3y\) is \(3\).
• The coefficient of \(-y\) is \(-1\).

By doing the arithmetic on the coefficients \(3 - 1\), we arrive at \(2y\). This process of combining like terms helps in reducing the complexity of an expression, making it easier to handle and understand.

Simplifying Expressions

Simplifying expressions is a fundamental step in solving algebraic problems. It means reducing an expression to its simplest form without changing its value. This often involves removing parentheses and combining like terms, as well as using basic arithmetic operations.

The distributive property is very helpful when simplifying. It lets us distribute a single term over terms inside the parentheses. However, in our original example \(3y - y\), the terms are already separate, so we directly combine them using the like term technique.

Remember, when simplifying expressions:

• Look for like terms.
• Combine them using their coefficients.
• Ensure that there is no further simplification possible.

For our expression \(3y - y\), combining the terms as described shows the expression is simplified to \(2y\). This final expression is clearer and more straightforward than the original.

Prealgebra Concepts

Prealgebra lays the groundwork for understanding algebraic concepts by focusing on arithmetic and basic mathematical principles. This stage involves becoming comfortable with numbers, operations, variables, and basic properties such as the distributive property.

In prealgebra, recognizing patterns and understanding properties such as the distributive property is crucial. It states: \[a(b + c) = ab + ac\]This property is useful not only when multiplying but also when simplifying expressions and ensuring terms can be added or subtracted efficiently.

The expression \(3y - y\) is a simple example that shows how prealgebra concepts can be applied to understand more complex algebraic structures. By extracting the \(y\) and simplifying the coefficients, students see the direct results of combining like terms and employing the distributive property, setting the stage for future algebraic success.