Problem 69 - Prealgebra

Question

Simplify each of the following expressions as much as possible. $$3(y-5)+6$$

Step-by-Step Solution

Answer

The expression simplifies to \(3y - 9\).

Step 1: Distribute the 3 into the Parentheses

The expression given is \(3(y-5) + 6\). The first step is to use the distributive property to eliminate the parentheses by multiplying the 3 by each term inside the parentheses. This gives: \(3 \times y + 3 \times (-5) = 3y - 15\). After distributing, the expression becomes \(3y - 15 + 6\).

Step 2: Combine Like Terms

Now, simplify the expression by combining like terms. The terms \(-15\) and \(+6\) are both constants and can be combined. Thus, \(-15 + 6 = -9\). The simplified expression now becomes \(3y - 9\).

Key Concepts

Distributive Property

The distributive property is a key concept in algebra that helps simplify expressions involving parentheses. When you have an expression like \(3(y-5) + 6\), the distributive property allows you to remove the parentheses by distributing, or multiplying, a number outside the parentheses by each term inside.It's like sharing a pie among several people. Each person gets an equal slice, just as each term inside the parentheses gets multiplied by the number outside. In our expression:

• Multiply 3 by \(y\) to get \(3y\).
• Multiply 3 by \(-5\) to get \(-15\).

After using the distributive property, the expression \(3(y-5) + 6\) becomes \(3y - 15 + 6\). This step sets up the expression for the next stage of simplification by removing the parentheses.

Combining Like Terms

Once the distributive property has been used to simplify the expression, the next step is to combine like terms. This means to merge terms in an expression that have the same variables raised to the same power, such as combining constant numbers or terms with the same variable.In the expression \(3y - 15 + 6\), you need to combine the constants \(-15\) and \(6\). These are the numbers without variables, and they can be simplified as follows:

• Calculate \(-15 + 6 = -9\).

After combining these like terms, the expression becomes \(3y - 9\). It's often helpful to write down each step clearly to avoid mistakes and ensure that you've properly simplified the expression.

Prealgebra

Prealgebra is the foundation of mathematics that prepares students for the more advanced concepts of algebra. Simplifying expressions, such as the one seen in this exercise, is a fundamental skill taught in prealgebra.In prealgebra, you learn to:

• Understand and apply properties of operations, such as the distributive property.
• Identify and combine like terms to simplify expressions.
• Perform basic calculations with integers and variables.

By mastering these skills, you build a strong mathematical foundation. Simplifying expressions, such as \(3(y-5) + 6\), not only helps you develop problem-solving skills but also prepares you for the challenges you will encounter in algebra and beyond.