Problem 85 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ -4(2 x-3 y) $$

Step-by-Step Solution

Answer

Question: Simplify the expression -4(2x - 3y). Answer: The simplified expression is -8x + 12y.

Step 1: Apply the Distributive Property

Multiply -4 by both terms of the binomial \((2x - 3y)\): $$ -4(2x - 3y) = -4 \cdot 2x + (-4) \cdot (-3)y $$

Step 2: Perform the Multiplication

Multiply the numbers together: $$ -4 \cdot 2x + (-4) \cdot (-3)y = -8x + 12y $$

Step 3: Write the Simplified Expression

The given expression is now simplified. The simplified expression is: $$ -8x + 12y $$

Key Concepts

Distributive Property

When we talk about simplifying algebraic expressions, one crucial technique often comes into play: the distributive property. It's a basic arithmetic rule that allows us to multiply a single term by each term within a set of parentheses.

Consider the expression \( -4(2x - 3y) \). To simplify this, we distribute the \( -4 \) to both \( 2x \) and \( -3y \). This is essentially saying, \( -4 \) times \( 2x \) plus \( -4 \) times \( -3y \) gives us \( -8x + 12y \).

Remember, when multiplying two numbers with like signs (both positive or both negative), the result is positive, and when multiplying numbers with unlike signs (one positive, one negative), the result is negative. This rule helps avoid common mistakes when applying the distributive property to expressions involving subtraction and negative coefficients.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables (like \( x \) and \( y \)), and operation symbols (like \( + \) for addition and \( - \) for subtraction). They don't have an equals sign like equations do. In our exercise, \( -4(2x - 3y) \) is an algebraic expression.

An important part of working with algebraic expressions is understanding how to simplify them. Simplification might involve combining like terms, using the distributive property, or performing operations on numbers and variables. The main goal is to write the expression in its most reduced form to make it easier to understand or use in further calculations.

Binomial Multiplication

In algebra, we often multiply binomials, which are expressions that have two terms, such as \( (2x - 3y) \). When multiplying a monomial (a single term, like \( -4 \)) by a binomial, we use the distributive property. In more complex cases, such as multiplying two binomials, we use the FOIL method (First, Outer, Inner, Last) to multiply each term in the first binomial by each term in the second binomial.

In our simple example, we only multiply the monomial \( -4 \) with the binomial \( (2x - 3y) \), resulting in \( -8x + 12y \). Had there been another binomial instead of a monomial, we would've expanded the products accordingly to simplify the expression.

Elementary Algebra

Elementary algebra is the field of mathematics that introduces algebraic expressions, the use of variables to represent numbers, and the basic operations of algebra, such as simplifying expressions, solving equations, and factoring.

In the context of elementary algebra, understanding how to manipulate and simplify expressions is essential. The example \( -4(2x - 3y) \) involves applying fundamental concepts like the distributive property and performing multiplications that are core to elementary algebra. Mastering these basics enables students to tackle more complex problems and paves the way for advanced mathematical studies. Always ensure each step in simplifying an expression is clear and logically follows from the previous step, as this will help avoid errors and deepen your understanding of how algebra works.