Problem 37 - Algebra 1

Question

DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the expression without parentheses. $$ 2(3 x-1) $$

Step-by-Step Solution

Answer

The expression without parentheses will be \(6x - 2\).

Step 1: Identify Variables and Values

Identify the real numbers \(a\), \(b\), and \(c\) in the expression according to the distributive property. In this case, \(a = 2\), \(b = 3x\), and \(c = -1\).

Step 2: Multiply a and b

Multiply \(a\) by \(b\) to obtain \(ab\). In this case, it's \(2 * 3x\) which equals to \(6x\).

Step 3: Multiply a and c

Multiply \(a\) by \(c\) to obtain \(ac\). Here, \(a * c = 2 * -1\), which equals \(-2\)

Step 4: Rewrite Using Distributive Property

Add or subtract (according to the sign) the results \(ab\) and \(ac\) obtained in steps 2 and 3 to rewrite the expression without parentheses. So, \(2(3x - 1)\) equals to \(6x - 2\)

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. In this exercise, the expression is \(2(3x-1)\). Here, \(3x\) and \(-1\) are part of an expression inside parentheses, multiplied by \(2\). The goal is to simplify it using the distributive property without changing its value.

An algebraic expression can be simplified by applying mathematical operations systematically. This makes it easier to understand and solve further mathematical problems. By simplifying an expression, you might make complex calculations more manageable. It is beneficial for students to recognize terms and factors within an expression. Terms are separated by addition or subtraction, while multiplication or division connects factors.

Understanding the components of an algebraic expression, such as coefficients (the numerical part of a term with a variable like \(3\) in \(3x\)), and constants (fixed values like \(-1\)), helps in simplifying and solving the expression effectively.

Arithmetic Operations

Arithmetic operations include basic math operations such as addition, subtraction, multiplication, and division. Applying these operations correctly is key to solving algebraic expressions. In the expression \(2(3x-1)\), multiplication is the primary operation used.

The first arithmetic operation is the multiplication of the coefficient 2 with \(3x\), resulting in \(6x\). This shows how multiplication distributes over terms within parentheses. Next, the same coefficient is multiplied by \(-1\), producing the result \(-2\).

When tackling arithmetic operations, it is important to consider the order of operations, sometimes remembered through the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This ensures operations are applied in the correct order. In our example, handling parentheses and multiplication first follows these principles and leads to the correct simplification of the expression.

Mathematical Properties

The distributive property is a fundamental mathematical property applied in this exercise. It involves a number outside a parenthesis multiplying each term inside the parenthesis. For the expression \(2(3x - 1)\), you apply the distributive property by multiplying the \(2\) with both terms, \(3x\) and \(-1\).

This property is stated mathematically as \(a(b + c) = ab + ac\). It shows how multiplication is distributed over addition or subtraction. This property helps simplify expressions, making them easier to work with for further calculations.

Other mathematical properties include the commutative property (order doesn't matter for addition or multiplication) and associative property (grouping doesn't matter for addition or multiplication). However, the distributive property is uniquely valuable when dealing with algebraic expressions that involve parentheses, directly simplifying expressions and revealing equivalent forms.