Problem 84 - Beginning and Intermediate Algebra

Question

Use the distributive property to rewrite each expression. $$ -5.2(4 x+2.3) $$

Step-by-Step Solution

Answer

-20.8x - 11.96

Step 1: Identify the Distributive Property

The distributive property states that for any numbers a, b, and c, the equation a(b + c) = ab + ac holds. This means we can distribute the multiplication across the terms inside the parenthesis.

Step 2: Apply the Distributive Property

Multiply -5.2 by each term inside the parenthesis: \(-5.2(4x + 2.3) = -5.2 \cdot 4x + (-5.2) \cdot 2.3\)

Step 3: Perform the Multiplications

Now, multiply the coefficients: \(-5.2 \times 4 = -20.8\) and \(-5.2 \times 2.3 = -11.96\)

Step 4: Combine the Results

Combine the results to rewrite the expression: \(-20.8x - 11.96\)

Key Concepts

algebraic expressions

In algebra, expressions are combinations of numbers, variables, and operations. Unlike equations, they do not have an equal sign. An example is -5.2(4x + 2.3). Here, -5.2 is a coefficient, and 4x + 2.3 is an algebraic expression containing both a variable term and a constant term. It’s essential to understand that expressions can be simplified using algebraic rules. By applying these rules, you can manage and solve more complex problems efficiently. The distributive property is one of these key rules. It allows you to distribute multiplication across terms within parentheses and simplify the expression further.

multiplication in algebra

To understand multiplication in algebra, let’s consider the problem -5.2(4x + 2.3). Here, the term outside the parenthesis (-5.2) needs to be multiplied by each term inside the parenthesis (4x and 2.3). This process is known as distributing the multiplication. It's crucial to keep track of both the coefficients and the signs (positive or negative) during this process. For example, multiplying -5.2 by 4x gives -20.8x. Similarly, multiplying -5.2 by 2.3 results in -11.96. This step-by-step multiplication ensures that each term inside the parenthesis is accounted for properly, helping to simplify the expression fully.

coefficients

Coefficients are the numerical factors in terms of an algebraic expression. In -5.2(4x + 2.3), -5.2 and 4 are coefficients. Coefficients tell us how many times to count the variable. When simplifying expressions, you multiply the coefficients accordingly. For instance, in the example above, we multiply -5.2 by each term inside the parenthesis. -5.2 times 4x gives -20.8x. Similarly, multiplying -5.2 by 2.3 results in -11.96. Knowing how to handle coefficients correctly is crucial when working with algebraic expressions, ensuring accurate results.