Problem 49 - College Algebra with Modeling and Visualization

Question

Apply the distributive property. $$-2(5 x+9)$$

Step-by-Step Solution

Answer

The simplified expression is \(-10x - 18\).

Step 1: Identify the Constants and Variables

In the expression \[-2(5x + 9)\]we have a constant \(-2\), a variable \(x\), and another constant \(9\). Our task is to apply the distributive property to simplify this expression.

Step 2: Apply the Distributive Property to Each Term

The distributive property states that \[a(b + c) = ab + ac\]In this case, our 'a' is \(-2\), our 'b' is \(5x\), and our 'c' is \(9\). We will distribute \(-2\) to both \(5x\) and \(9\):1. Multiply \(-2\) by \(5x\):\[-2 \times 5x = -10x\]2. Multiply \(-2\) by \(9\):\[-2 \times 9 = -18\]

Step 3: Combine the Results

Combine the results from applying the distributive property: \[-10x - 18\]

Key Concepts

Constants

In mathematics, a constant is a fixed value that does not change. In an algebraic expression, constants are represented by numbers without any variables attached to them.

• For example, in the expression \(-2(5x + 9)\), \(-2\) and \(9\) are constants.
• They remain the same no matter what value is substituted for the variable.

Understanding constants is crucial because they are the stable pieces of the equation or expression. When you perform operations such as addition or multiplication, constants behave just like usual numbers. This means when you apply the distributive property, each constant should be treated as a separate, unchanging number.

Variables

Variables are symbols used to represent unknown or changing quantities in algebraic expressions. Typically, variables are denoted by letters such as \(x, y,\mathrm{ or } \,z\).

• In our example, \(5x\), the \(x\) is the variable.
• The variable can take on various values, allowing it to represent different outcomes in mathematical calculations and equations.

Grasping the concept of a variable is essential because it forms the foundation of algebra. Variables allow us to form relationships between numbers and symbols, explore patterns, and find unknown values.

Algebraic Expressions

An algebraic expression is a combination of constants, variables, and operations (such as addition, subtraction, multiplication, and division). Unlike equations, algebraic expressions do not contain an equality sign.

• An example of an algebraic expression is \(-2(5x + 9)\).
• It includes a constant \(-2\), a numerical coefficient \(5\) attached to the variable \(x\), and another constant \(9\).

Algebraic expressions can be simplified using various properties and operations, like the distributive property, to make them easier to work with or solve.

Simplifying Expressions

Simplifying an expression means transforming it into a simpler, more compact form while keeping its value unchanged. This process often involves applying mathematical properties such as the distributive property.

• In the expression \(-2(5x + 9)\), simplifying involves distributing \(-2\) to both \(5x\) and \(9\).
• You perform the operations: \(-2 \times 5x = -10x\) and \(-2 \times 9 = -18\).
• The final simplified form is \(-10x - 18\).

Simplifying expressions makes them more manageable and often paves the way for solving more complex mathematical problems. It highlights the relationships between the terms clearly, which is particularly useful in solving algebraic equations.