Problem 43 - Prealgebra

Question

Use the Distributive Property to write each expression as an equivalent algebraic expression. $$-2(z-4)$$

Step-by-Step Solution

Answer

The equivalent expression is \(-2z + 8\).

Step 1: Identify the Expression

The given expression is \(-2(z-4)\).

Step 2: Apply the Distributive Property

The distributive property states that \(a(b+c) = ab + ac\). Apply it here: distribute the \(-2\) to both \(z\) and \(-4\).\[-2(z - 4) = (-2) \cdot z + (-2) \cdot (-4)\]

Step 3: Multiply Each Term

Multiply \(-2\) and \(z\): \((-2) \times z = -2z\).Next, multiply \(-2\) and \(-4\): \((-2) \times (-4) = 8\).

Step 4: Form the Equivalent Expression

Combine the results from step 3 to form the equivalent expression:\(-2z + 8\).

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental part of algebra, consisting of numbers, variables, and operations put together to represent a mathematical idea. An algebraic expression is like a sentence that can include:

• Variables, such as \(z\) or \(x\), which stand in for unknown numbers.
• Constants, like \(2\) or \(-4\), which have a fixed value.
• Operators, such as addition (+), subtraction (−), multiplication (×), and division (÷).

In the expression \(-2(z-4)\), \(z\) is the variable, under operations between \(-2\) and \(-4\). Understanding how these parts interact is essential to forming correct equations. Algebraic expressions can be simplified into more manageable forms, which we explore further in the next sections.

Equivalent Expressions

Equivalent expressions are different algebraic expressions that simplify to the same value for any substitution of their variables. If you take two expressions and, after simplifying, they represent the same quantity, they are equivalent.
For example, with the expression \(-2(z-4)\), after applying the distributive property and simplifying, we find that it can be rewritten as \(-2z + 8\). Despite their different appearances, both expressions will yield the same result for any value of \(z\). Thus, they are equivalent.
Checking equivalence often involves:

• Using mathematical properties, like the distributive property, to expand or factor expressions.
• Simplifying expressions by combining like terms or reducing fractions.
• Substituting specific values for variables to verify that two expressions have the same outcome.

This concept is important because it allows you to transform complex expressions into simpler forms without changing their value.

Multiplication in Algebra

Multiplication in algebra involves applying the same rules of multiplication you use with numbers but includes variables. It is essential for simplifying and manipulating algebraic expressions.
In our example \(-2(z-4)\), multiplication is used as follows:

• Distribute the number outside the parenthesis across each term inside: \(-2\times z\) and \(-2\times-4\).
• The result is that each term in the parenthesis is multiplied individually, allowing the expression to expand.

This process shows how multiplication works alongside algebraic rules, such as the distributive property, to simplify or rearrange expressions.
Additionally, understanding multiplication helps you solve equations by isolating variables, which is foundational in algebra. Correctly applying multiplication ensures you maintain balance in an equation or algebraic expression, advancing your ability to analyze and solve for unknowns effectively.