Problem 34 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ 2(x-10) $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression: 2(x - 10) Answer: The simplified expression is 2x - 20.

Step 1: Apply the distributive property of multiplication.

To apply the distributive property, we multiply the number outside the parentheses with each term inside the parentheses. So we have: $$ 2(x - 10) = 2 \cdot x - 2 \cdot 10 $$

Step 2: Perform the multiplication.

Now, we have to perform the multiplication to simplify the expression: $$ 2 \cdot x - 2 \cdot 10 = 2x - 20 $$ In this case, we don't have any like terms to combine, so the simplified expression is: $$ 2x - 20 $$

Key Concepts

Multiplying Binomials

When multiplying binomials, the distributive property is your best friend. A binomial is a two-term algebraic expression, and in many cases, both involve a simple variable multiplied by a coefficient, like \( x - 10 \). To multiply a binomial by a number—as shown in our problem, \( 2(x - 10) \)—you distribute the multiplier across each term inside the parentheses.

Here's how you do it: apply the distributive rule, which involves multiplying the number outside the parentheses by each term within the binomial. In this case, it means calculating \( 2 \times x \) and \( 2 \times (-10) \). This operation effectively helps "spread" the multiplier over each part of the binomial.

Understanding the steps of multiplication with binomials not only simplifies the expression but also sets the stage for further algebraic manipulation. Applying these principles to various problems will boost your confidence with algebraic expressions.

Simplifying Expressions

The essence of simplifying expressions is to make complex algebraic statements easier to work with. After distributing the terms in the binomial, you need to perform the multiplication.

In our example, once we distribute \( 2 \) across the terms in \( x - 10 \), the expression \( 2(x - 10) \) becomes \( 2x - 20 \). This step requires careful execution to avoid mistakes. Multiplying \( 2 \) by \( x \) yields \( 2x \), while \( 2 \) times \(-10\) gives you \(-20\).

This multiplication is vital because it converts the expanded expression into its simplest form without changing its value, ensuring it aligns with the foundational operations in algebra.

Combining Like Terms

After simplifying expressions, the next potential step is combining like terms. "Like terms" are components of the expression that have identical variable parts raised to the same power. They can be seamlessly combined by adding or subtracting coefficients.

In our example, \( 2x - 20 \), there are no like terms present. "Like" refers to the coefficient (here \( 2 \) with \( x \)), that might exist with like terms elsewhere in more complex expressions.

While combining like terms isn't necessary here, understanding how to identify them is critical for more complex algebraic problems. When present, combining them reduces the expression's complexity, helping you find solutions more easily and efficiently. Handling different situations with like terms strengthens algebraic foundation skills.