Question

Use the distributive property and mental math to simplify the expression. $$ -3 y-2 x $$

Short answer

The simplified expression is -3y - 2x.

Step-by-step solution

Recognize the structure of the expression

The expression given is -3y - 2x. There are no parenthetical expressions, so there's no opportunity to apply the distributive property.

Use mental math

You may be tempted to use the distributive property; however, due to the lack of parenthetical expressions, no distribution is needed. The expression -3y - 2x is already the simplest expression representing these variables.

Final Answer

There's nothing more to simplify, so the original expression, -3y - 2x, is the final simplified expression.

Key concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to simplify expressions and perform multiplication over addition or subtraction within parentheses. The property is expressed as:
\[ a \times (b + c) = a \times b + a \times c \]
and similarly for subtraction:
\[ a \times (b - c) = a \times b - a \times c \]
This property can be particularly useful when we wish to eliminate parentheses in an algebraic expression. However, in the given exercise with the expression \(-3 y-2 x\), no parentheses are present, meaning the distributive property does not come into play. Understanding when to use it, and just as importantly, when not to, is crucial to simplifying expressions correctly.

Mental Math

Mental math involves performing arithmetic calculations without the aid of a calculator or paper. It's a skill that can greatly speed up the process of algebraic manipulations by allowing quick computations and simplifications in your head. To master mental math, one must be familiar with the basic arithmetic operations and their properties.
When looking at the expression \(-3 y - 2 x\), mental math tells us there are no like terms to combine, and with no further operations applicable, it already stands as the simplest form. The absence of any parentheses means we simply recognize the expression's terms and know that they cannot be added or subtracted from each other since they're not algebraically alike.

Algebraic Structure

Algebraic structure refers to the way algebraic terms are arranged and related to each other within an expression. It encompasses understanding variables, coefficients, constants, operations and the underlying rules that govern them. For the expression given, \(-3y - 2x\), we have two terms, \(-3y\) and \(-2x\), each consisting of a variable and a coefficient. The structure shown is simple; it indicates subtraction between these non-like terms. Recognizing this algebraic structure is critical, as it informs us that the terms cannot be simplified through addition or subtraction. Familiarity with different algebraic structures aids in identifying the applicable arithmetic operations to simplify expressions correctly.