Problem 43 - Prealgebra

Question

Use the Distributive Property to rewriteh expression as an equivalent algebraic expression. \(2(1.2 c+14)\)

Step-by-Step Solution

Answer

The equivalent expression is \(2.4c + 28\).

Step 1: Understand the Distributive Property

The Distributive Property states that for all real numbers \(a\), \(b\), and \(c\), \(a(b+c) = ab + ac\). This property allows us to multiply a single term by each term inside a parenthesis and add the results.

Step 2: Identify terms to distribute

The expression \(2(1.2c + 14)\) consists of the number \(2\) being multiplied by the expression inside the parentheses \((1.2c + 14)\). According to the Distributive Property, we need to distribute \(2\) to each term inside the parenthesis: \(1.2c\) and \(14\).

Step 3: Apply the Distributive Property

Multiply \(2\) by each term inside the parentheses: - First, multiply \(2\) by \(1.2c\) to get \(2 \times 1.2c = 2.4c\).- Next, multiply \(2\) by \(14\) to get \(2 \times 14 = 28\).

Step 4: Write the equivalent expression

Combine the results from the previous step into an expression: The equivalent expression is \(2.4c + 28\).

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. They are used to represent mathematical relationships in a simplified form. In our example, the expression given is \(2(1.2c + 14)\). Here, \(1.2c + 14\) is an algebraic expression inside the parentheses, where \(c\) is a variable that can take different values. This expression is being multiplied by 2, which is outside the parentheses.
Algebraic expressions can be manipulated using various properties and operations to create new, equivalent expressions or to simplify complex expressions. By understanding how these expressions work, you can solve equations and identify relationships more easily.

• Variables like \(c\) are symbols that can represent any number.
• The coefficients (numbers in front of the variables, such as \(1.2\) in \(1.2c\)) indicate how much the variable is multiplied by.
• Constants like \(14\) are fixed numbers that don't change.

Understanding these components helps in identifying the correct application of mathematical properties like the Distributive Property.

Multiplication

Multiplication is one of the basic arithmetic operations where two quantities, known as factors, are combined to produce a product. In our problem, multiplication is used between the number 2 and the expression inside the parentheses \(1.2c + 14\).
When distributing, you multiply the number outside the parentheses by each term inside the parentheses separately. This means every term inside is treated as an individual multiplication problem.

• \(2 \times 1.2c\): Here, you multiply \(2\) by \(1.2\) resulting in \(2.4\), and the variable \(c\) remains with it, giving you \(2.4c\).
• \(2 \times 14\): This is simply multiplying the constants, resulting in \(28\).

Being skilled in multiplication can greatly aid in simplifying expressions and solving equations.

Equivalent Expressions

Equivalent expressions are different expressions that have the same value for all values of the variables within them. When you apply mathematical operations like the Distributive Property to an algebraic expression, you can rewrite it in different forms that are still equivalent.
For example, the expression \(2(1.2c + 14)\) was rewritten using the Distributive Property to become \(2.4c + 28\). Even though they look different, both of these expressions represent the same mathematical quantity.
It is important to understand the concept of equivalent expressions for several reasons:

• They allow you to simplify expressions, making them easier to work with, particularly in solving problems.
• Just because expressions are equal, they may not look similar. Recognizing equivalence can be important in verifying solutions to algebraic problems.

Mastering the skill of finding and understanding equivalent expressions can enhance your problem-solving abilities in algebra and beyond.