Problem 42 - Prealgebra

Question

Use the Distributive Property to write each expression as an equivalent algebraic expression. $$(x-3) 12$$

Step-by-Step Solution

Answer

The expression \((x-3)12\) simplifies to \(12x - 36\).

Step 1: Identify the Terms

The expression given is \((x-3) \cdot 12\). According to the distributive property, we need to distribute the multiplier, which is 12, to each term inside the parentheses.

Step 2: Distribute the Multiplier

Apply the distributive property by multiplying each term inside the parentheses by 12: \(x \times 12\) and \(-3 \times 12\).

Step 3: Calculate the Products

First, calculate \(x \times 12 = 12x\). Then, calculate \(-3 \times 12 = -36\).

Step 4: Combine the Results

Combine the results of the multiplication to form the equivalent expression: \(12x - 36\).

Key Concepts

Algebraic Expression

Algebraic expressions are combinations of letters and numbers that represent a mathematical idea or relationship. They are like sentences in mathematics where letters, known as variables, stand in for unknown values. In our example,

• The expression \((x - 3) \cdot 12\) is an algebraic expression.
• Here, \(x\) is the variable and -3 and 12 are constants.
• Variables allow expressions to represent general formulas and principles, not just specific numbers.

These expressions can include operations such as addition, subtraction, multiplication, and division. The power of algebraic expressions lies in their ability to model real-world scenarios and solve for unknowns. By manipulating these expressions using algebraic rules, like the distributive property, we can simplify or rearrange equations to find solutions or equivalent forms.

Multiplication

Multiplication in algebra operates under similar principles to basic arithmetic but extends to include variables. When dealing with algebraic expressions,

• You multiply constants just like numeric multiplication.
• When multiplying variables or a mix of constants and variables, each term gets individually multiplied.

When applying the distributive property, as shown in the exercise,

• You distribute the multiplier - here it's 12 - to each term inside the parentheses.
• This means calculating \(x \times 12\) and then \(-3 \times 12\).

This approach helps in expanding the expression, making it easier to work with or solve depending on what you need. Multiplication builds the foundation for many other algebraic operations that follow, like combining like terms or solving equations.

Equivalent Expressions

Equivalent expressions are different mathematical statements that express the same quantity. They might look different, but when simplified or evaluated, they give the same result. For instance,

• The original expression \((x - 3) \cdot 12\) simplifies to the equivalent expression \(12x - 36\).
• These two expressions represent the same relationship, but the latter does not use parentheses, which can often simplify further computations.

Finding equivalent expressions is crucial because it allows us to restate problems in simpler or more useful forms. It can help in reducing complexity or in solving for variables. Using properties like the distributive property is a common method to find such equivalent expressions in algebra. This enables better manipulation of algebraic equations during problem-solving.