Problem 40 - Prealgebra

Question

Apply the distributive property to each expression and then simplify. $$5(2 y-6)+4 y$$

Step-by-Step Solution

Answer

The simplified expression is \(14y - 30\).

Step 1: Apply the Distributive Property

Use the distributive property to expand the expression \(5(2y - 6)\). Distribute the 5 to both terms inside the parentheses: \[5 imes 2y + 5 imes (-6) = 10y - 30\]

Step 2: Combine Like Terms

Now, take the expanded expression from Step 1, which is \(10y - 30\) and combine it with \(+4y\) from the original expression. You combine like terms, which are \(10y\) and \(4y\): \[10y + 4y - 30 = 14y - 30\]

Step 3: Final Simplified Expression

After combining like terms, the final simplified expression is \(14y - 30\). This is the expression in its simplest form and there are no like terms left to combine.

Key Concepts

Combining Like Terms

In algebra, combining like terms is an important step that helps us simplify expressions. If you see terms that have the same variable raised to the same power, these can be considered as ​like terms​. By adding or subtracting these terms, you reduce the complexity of the expression.

Let's look at the exercise: after applying the distributive property, we obtained the expression \(10y - 30 + 4y\). Here, the terms \(10y\) and \(4y\) are like terms because they both have the variable \(y\). So, you add them to get \(14y\). The number \(-30\) doesn't have a \(y\) attached to it, so it stays as it is in this step.

• Always look for terms with the same variable part.
• Add or subtract the coefficients of these like terms.
• Maintain the variable during the operation.

By focusing on combining like terms, you simplify algebraic expressions step by step, making them easier to handle in solving equations.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This process not only makes the expression easier to work with but also clarifies the relationships between different terms.

In our example, we started with the expression \(5(2y-6)+4y\). Using the distributive property allowed us to expand this, and combining like terms led us to a more simplified form: \(14y-30\).

To simplify an expression, follow these steps:

• Expand the expression using properties like the distributive property.
• Combine like terms to condense the expression.
• Make sure there are no terms left to combine or simplify further.

It's like tidying up a messy room — everything is clearer and more organized. Simplifying expressions is crucial for further algebraic operations, such as solving equations or making substitutions.

Pre-Algebra

Pre-algebra is all about laying the foundation for future algebra courses. The focus is on understanding concepts such as variables, expressions, and fundamental operations.

Working with expressions like \(5(2y-6)+4y\) in pre-algebra teaches you several essential skills:

• Understanding the role of variables and coefficients.
• Learning to apply mathematical properties, such as the distributive property.
• Developing skills to simplify expressions and solve simple equations.

These skills are foundational and you continue to build on them as you progress into more complex algebraic and mathematical concepts. Pre-algebra focuses on mastering these basics, which makes every subsequent step in math more approachable.