Problem 60 - 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. $$ -11(5 x+3)+10 $$

Step-by-Step Solution

Answer

-55x - 23

Step 1: Apply the Distributive Property

The distributive property allows us to multiply a single term by each term inside the parentheses. Here, you need to multiply \(-11\) by each term inside the parentheses \((5x+3)\). Thus, it can be rewritten as: \[-11(5x) + (-11)(3)\].

Step 2: Perform the Multiplication

Now, perform the multiplication for each term. First, multiply \(-11\) and \(5x\) to get \(-55x\). Next, multiply \(-11\) and \(3\) to get \(-33\). The expression now becomes \(-55x - 33\).

Step 3: Add the Constant Term

Now you need to add the constant term outside the brackets, which is \(+10\), to the simplified expression from the previous step. This gives us: \(-55x - 33 + 10\).

Step 4: Simplify the Expression

Combine like terms by performing arithmetic with the constant terms \(-33 + 10\), which simplifies to \(-23\). The expression is now \(-55x - 23\).

Key Concepts

Understanding Expressions

Expressions are a combination of numbers, variables, and operations that represent a particular value. In algebra, understanding expressions is crucial because it helps in solving equations, modeling real-world problems, and understanding relationships between variables. For example, in the exercise provided, the expression \(-11(5x+3)+10\) uses multiplication, addition, and includes a variable, which is a letter that represents an unknown value.

• The expression has two parts inside the parentheses, \(5x\) (which is called a term with a variable) and \(3\) (which is a constant term).
• Outside the parentheses, \(-11\) acts as a coefficient, meaning it multiplies both terms in the brackets.
• This expression also has a \(+10\), which is a constant term added separately to the result of the distribution.

By understanding these basic components of an expression, you can more easily apply algebraic operations like the distributive property to simplify it.

Simplification Process

Simplification is the process of making an algebraic expression easier to work with by combining like terms and performing operations. It helps in understanding the essence of the expression without changing its value. Applying the distributive property is a fundamental simplification technique.

When using the distributive property, you multiply the term outside the parentheses by each term inside. In the provided problem, this involves:

• Multiplying \(-11\) by \(5x\) to get \(-55x\).
• Multiplying \(-11\) by \(3\) to get \(-33\).

The simplified expression becomes \(-55x - 33\).

The next step is combining this result with the constant term \(+10\) by performing arithmetic operations. Simplifying further, \(-33 + 10\) results in \(-23\), leading to the final simplified expression \(-55x - 23\). By following these steps, you make the expression neater and more understandable.

The Role of Constant Terms

Constant terms in an algebraic expression are numbers without variables, and they remain unchanged through most operations. They play a vital role in simplifying expressions by being directly combined or adjusted through arithmetic operations.

In the exercise, we encounter constant terms at different points:

• The constant \(3\) inside the parentheses gets directly multiplied by the \(-11\) from outside, resulting in \(-33\).
• Another constant, \(+10\), is waiting outside the parentheses and is added after simplification.

By combining these constant terms, \(-33\) and \(+10\), we obtain \(-23\). This completes our simplification of the constant terms.

Through constant terms, we bridge the variable-oriented part of expressions, helping to translate and simplify the effects of operations while preserving the overall balance. They are essential to reaching the simplest form of an expression.