Problem 49 - Algebra 1: Concepts and Skills

Question

Use the distributive property to rewrite the expression without parentheses. $$ -(6 y-5) $$

Step-by-Step Solution

Answer

The expression \[-(6y-5)\] without parentheses is \[-6y+5\].

Step 1: Distribute the coefficient

In this case, the outer term is -1 because of the negative sign. Multiply each term inside the parentheses by -1: \[-1 \cdot 6y\] and \[-1\cdot -5\].

Step 2: Multiply

After multiplying, you get: \[-6y\] and \[+5\].

Step 3: Write the final answer

The expression \[-(6y-5)\] without parentheses is \[-6y+5\].

Key Concepts

Understanding Negative Numbers

We often encounter negative numbers when dealing with real-life situations, such as temperatures below zero or owing money, which are depicted as negative values in a bank account. In math, a negative number is simply a number less than zero. When using negative numbers:

• If you multiply two negative numbers, the result is positive.
• If you multiply a negative number by a positive one, the result is negative.
• Adding negative numbers is like subtracting: for example, adding a negative number is the same as moving to the left on a number line.

Understanding negative numbers is crucial for working with algebraic expressions, especially when using the distributive property, as you need to manage the sign of each term carefully.
In our example, when you distribute the egative sign across a set of parenthesis, each term inside the parenthesis becomes negative if it was initially positive and positive if it was originally negative.

Algebraic Expressions Explained

Algebraic expressions combine numbers, variables, and operations. Think of them as mathematical phrases that can represent real-world problems. They're constructed using constants (like numbers 5, 10), variables (like x, y—which represent unknown values), and operations such as addition ( +), subtraction ( -), multiplication ( ×), and division ( ÷).
For example, an expression like 6y mixes a number with a variable, meaning 6 times whatever value ‘y’ represents.
In applying the distributive property, you multiply everything in parentheses by a factor outside. For instance, in the expression -(6y - 5), the negative sign acts as a multiplier across every term inside the parentheses, effectively applying to both 6y and -5. Getting familiar with writing, reading, and manipulating algebraic expressions leads to proficiency in solving equations and understanding complex mathematical concepts.

Simplifying Expressions Made Easy

Simplifying expressions is all about making them easier to work with or understand by eliminating unnecessary parts. This often involves reducing the complexity of polynomial expressions or eliminating parentheses using properties like the distributive property. When you simplify expressions, you combine like terms and ensure constants and variables are organized effectively. For instance, in the original exercise -(6y - 5), we use the distributive property to multiply -1 by each term inside the parentheses:

• The first multiplication: -1 × 6y gives us -6y,
• and the second: -1 × -5 becomes +5.

After these operations, the expression simplifies to -6y + 5, making it more straightforward for evaluations or comparisons in algebraic problems. Mastery of simplifying expressions is foundational for tackling more advanced mathematical challenges successfully.