Problem 71 - Algebra 1: Concepts and Skills

Question

Use the distributive property to rewrite the expression without parentheses. $$ -5(y+4) $$

Step-by-Step Solution

Answer

-5y - 20

Step 1: Identify the expression

The expression that we need to distribute is -5 over \(y + 4\). This means that -5 needs to multiply every element inside the parentheses.

Step 2: Apply the distributive property

The distributive property states that \(a(b + c) = ab + ac\). So, we use this property to distribute -5 over \(y + 4\), it gives \(-5 \cdot y + (-5) \cdot 4\).

Step 3: Simplify the expression

After multiplying, we get \(-5y - 20\).

Key Concepts

Simplifying Algebraic Expressions

Understanding how to simplify algebraic expressions is a fundamental skill in algebra. It involves reducing an expression to its simplest form, making it easier to work with. An expression might look complicated at first, but with the right methods, it can be broken down into a more manageable state. Simplification can involve several steps, including applying the distributive property, combining like terms, and reducing fractions.

For example, consider the expression \( -5(y+4) \) given in the exercise. To simplify this, you'll need to eliminate the parentheses, which is where the distributive property comes into play. By following the subsequent steps—distributing the -5 to both \( y \) and \( 4 \) and then combining any like terms—you're left with a much simpler expression, \( -5y - 20 \). Always ensure each part of the expression is addressed; none should be left behind. This practice will guide you to a correct and simplified answer.

Applying the Distributive Property

The distributive property is a cornerstone concept in algebra that allows you to simplify expressions within parentheses by distributing a single term across each term inside. The formal definition of the property is \( a(b + c) = ab + ac \), which simply means that you multiply the term outside the parentheses (\( a \)) by each term inside (\( b \) and \( c \) here).

When applied to our example, \( -5(y+4) \), we distribute the \( -5 \) across both \( y \) and \( 4 \) inside the parentheses, multiplying each by \( -5 \) to give \( -5 \) times \( y \) and \( -5 \) times \( 4 \). This yields \( -5y \) and \( -20 \) when simplified. It's essential to pay careful attention to the signs during distribution to avoid common mistakes, especially when negative numbers are involved.

Combining Like Terms

Once you've applied the distributive property, you might end up with an expression that has terms which can be combined. This is known as combining like terms. Like terms are terms that have exactly the same variable parts, and they can be added or subtracted from one another. When combining like terms, we conserve the variable part and simply add or subtract the numerical coefficients.

For instance, if after applying the distributive property, you get an expression such as \( -5y - 20 + 3y \), you would then combine the terms with \( y \) by adding their coefficients (\( -5 \) and \( 3 \) in this case), resulting in \( -2y - 20 \). Notice that the term \( -20 \) does not combine with the others since it doesn't have the same variable. Recognizing like terms and properly combining them is vital for achieving the correct simplified form of an expression.