Problem 40 - Algebra A Combined Function

Question

Use the distributive property to write each expression without parentheses. Then simplify the result, if possible. See Examples 7 through 12. $$ 2(x+5) $$

Step-by-Step Solution

Answer

2x + 10

Step 1: Apply the Distributive Property

The distributive property states that for any numbers \(a\), \(b\), and \(c\), the equation \(a(b+c) = ab + ac\) holds true. For this problem, apply the distributive property by distributing the 2 across the terms inside the parentheses: \[ 2(x + 5) = 2 \cdot x + 2 \cdot 5. \]

Step 2: Perform the Multiplications

Now compute the multiplications from Step 1: \(2 \cdot x\) remains \(2x\), and \(2 \cdot 5\) equals 10. Substitute these results back into the expression: \[ 2x + 10. \]

Step 3: Simplify the Expression

Examine the expression \(2x + 10\). Since \(2x\) and 10 do not have any common factors and are not like terms (one is a coefficient of \(x\) and the other is a constant), the expression is already simplified. Thus, \[ 2x + 10 \] is the simplified form.

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They are the building blocks of algebra. In the example provided, the expression \(2(x + 5)\) is an algebraic expression. It contains:

• A constant number 2 outside the parentheses
• A variable \(x\)
• A constant inside the parentheses, which is 5
• An operation symbol +, which denotes addition

These components come together to form the complete expression. Algebraic expressions are the basis for solving equations and understanding more complex mathematical problems. They can be manipulated using various arithmetic operations to achieve simplicity and clarity.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form while ensuring they are still equivalent to the original expression. This process often makes them easier to work with and understand.A common method of simplifying expressions is through the distributive property. In our original problem, we started with the expression \(2(x + 5)\). By applying the distributive property, we break it down to \(2 \cdot x + 2 \cdot 5\), resulting in \(2x + 10\).Once we've done this, we look for opportunities to combine like terms or further simplify each component. In this example, since \(2x\) and 10 are not like terms, the expression \(2x + 10\) is already in its simplest form.

Like Terms

Like terms are terms that contain the same variables raised to the same power. They are crucial in the simplification process because they can be combined to create a more streamlined expression. In our exercise, this concept helps us understand why \(2x\) and 10 cannot be combined.Let's break it down:

• \(2x\) is a term that involves the variable \(x\)
• 10 is a constant term without any variables

For terms to be "like," they need to have not just the same variable, but the variable must also be raised to the same power. Since \(2x\) has an \(x\) and 10 does not, they are not like terms and thus cannot be combined. Recognizing like terms is essential for successfully simplifying algebraic expressions.