Problem 30 - Algebra 1

Question

Simplify the variable expression. $$(-3)(-y)$$

Step-by-Step Solution

Answer

The simplified expression is \(3y\).

Step 1: Apply the multiplication rule for signs

When multiplying two negative values, the result is positive: \((-)(-)=(+)\).

\((-3)(-y) = (+3)(y) = 3y\)

Step 2: State the simplified result

The simplified expression is \(\boxed{3y}\).

Key Concepts

Multiplication of Negative Numbers

When multiplying two negative numbers, it might seem a bit puzzling at first. However, there is a straightforward rule to remember:

• The multiplication of two negative numbers results in a positive product.

This rule stems from the properties of numbers and how multiplication is defined. If you think about the number line, a negative number can be considered as going in the opposite direction. Multiplying two negative numbers flips the direction twice, bringing it back to the positive direction. For example, when multiplying \(-3\) and \(-y\), the negatives cancel each other out, resulting in a positive \(3y\). Understanding this basic rule will help simplify many algebraic expressions that involve negative numbers.

Variable Expressions

Variable expressions include numbers, variables (like \(x, y, z\)), and operations such as addition, subtraction, multiplication, or division. In our example, \((-3)(-y)\) is a variable expression.
Here are some important points about variable expressions:

• Variables are placeholders for quantities that can change.
• They allow you to form and solve equations based on general rules rather than specific numbers.

Working with variable expressions requires understanding how to combine these quantities using mathematical operations, which can lead to simplifying or solving them. For instance, the expression \((-3)(-y)\) uses a variable \(y\) and shows multiplication by \(-3\). Simplifying this leads to \(3y\), using the rules of negative number multiplication.

Algebraic Simplification

Algebraic simplification involves reducing an expression into its simplest form. This may include combining like terms, using rules of arithmetic, such as what we've done here.
To simplify an expression like \((-3)(-y)\), follow these steps:

• Identify and apply mathematical rules (e.g., multiplication rules for negative numbers).
• Combine like terms, if applicable.
• Rewrite the expression in a simpler form.

Through simplification, you'll make the expression easier to interpret and work with. This process is essential in solving equations, verifying solutions, and understanding the relationships between different parts of an equation. In our case, proper simplification of \((-3)(-y)\) leads to \(3y\), which is the simplest form of the original expression.