Problem 61 - Prealgebra

Question

Simplify each expression. $$-5(y+3)$$

Step-by-Step Solution

Answer

The simplified expression is \(-5y - 15\).

Step 1: Apply the Distributive Property

To simplify the expression \(-5(y+3)\), use the Distributive Property of multiplication over addition, which states that \(a(b+c) = ab + ac\). This means you multiply each term inside the parentheses by the term outside. Here, multiply \(-5\) by \(y\) and \(-5\) by \(3\).

Step 2: Perform Multiplication

Calculate each multiplication: \(-5 \times y = -5y\) and \(-5 \times 3 = -15\). Write down the results to form the expression: \(-5y - 15\).

Step 3: Combine the Like Terms

Since there are no like terms besides what we have simplified, \(-5y - 15\) is already in its simplest form and cannot be simplified further.

Key Concepts

Distributive Property

The distributive property is a foundational principle in algebra that helps simplify expressions, making complex problems more manageable. When dealing with an expression where a term is multiplied by a sum, such as \(-5(y+3)\), we use the distributive property. The property is expressed as follows: for any numbers \(a\), \(b\), and \(c\), the expression \(a(b+c)\) equals \(ab + ac\).
In our example:

• \(-5\) is the term outside the parentheses.
• \(y + 3\) represents the terms inside the parentheses.

We distribute \(-5\) to both \(y\) and \(3\), resulting in \(-5y - 15\). Overall, the distributive property simplifies multiplication problems and is a crucial skill for solving algebraic expressions.

Simplification

Simplification in algebra involves reducing expressions to their simplest form. This process not only makes expressions easier to understand, but it also prepares them for further operations. In our example, after applying the distributive property to \(-5(y+3)\), we ended up with the expression \(-5y - 15\).
Here's how simplification works:

• First, identify and perform basic operations, such as distribution or factoring.
• Then, rearrange or simplify terms to achieve the most concise expression possible.

In this case, there are no like terms to combine further. So, the simplest form of the expression remains \(-5y - 15\).Remember, simplification can involve other algebraic techniques, but the goal is always to streamline the equation or expression.

Basic Algebra

Basic algebra serves as the foundation for more advanced mathematical concepts. It involves operations such as addition, subtraction, multiplication, and division, using variables, constants, and sometimes parentheses. Understanding how to manipulate expressions using these operations is crucial:

• Variables, like \(y\) in our example, represent unknown values or quantities.
• Constants, such as \(-5\) and \(3\), are fixed values.
• Operations are applied carefully according to algebraic rules.

In the expression \(-5(y+3)\), basic algebra rules dictate that we perform multiplication, using the distributive property, before moving on to other operations. The goal of algebra is to create, manipulate, and simplify expressions and equations efficiently to find solutions or understand relationships between variables.