Problem 71 - Algebra A Combined Function

Question

Simplify each expression. See Section 1.8. $$ 2(x-5)+3(5-x) $$

Step-by-Step Solution

Answer

The expression simplifies to \(-x + 5\).

Step 1: Distribute the coefficients

Distribute the 2 in the term \(2(x-5)\) and the 3 in the term \(3(5-x)\). This means you will multiply each term inside the parentheses by the number outside: \[2(x-5) = 2 \cdot x - 2 \cdot 5 = 2x - 10\]\[3(5-x) = 3 \cdot 5 - 3 \cdot x = 15 - 3x\]

Step 2: Combine the distributed terms

Combine the two results from Step 1 into a single expression by writing them one after another with a plus sign in the middle: \[2x - 10 + 15 - 3x\]

Step 3: Group and combine like terms

Identify like terms in the expression and combine them. The like terms here are \(2x\) and \(-3x\), and the constant terms \(-10\) and \(15\): - For \(x\) terms: \(2x - 3x\) is \(-x\)- For constants: \(-10 + 15\) is \(5\)Combine these results to simplify the expression:\[-x + 5\].

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps us simplify expressions. It allows us to multiply a single term by each term inside a set of parentheses. This is essential when dealing with expressions like \(2(x-5)\) and \(3(5-x)\). By using the distributive property, you can break down complex expressions into simpler parts.

Here's how it works:

• Take the term outside the parentheses (in our case, 2 and 3) and multiply it with each term inside the parentheses.
• For \(2(x-5)\), multiply 2 by \(x\) to get \(2x\), and then multiply 2 by -5 to get -10, resulting in \(2x - 10\).
• For \(3(5-x)\), multiply 3 by 5 to get 15, and then multiply 3 by \(-x\) to get \(-3x\), resulting in \(15 - 3x\).

Using the distributive property makes handling variables and coefficients straightforward, preparing the expression for the next steps of simplification.

Combining Like Terms

After using the distributive property, the next step is to simplify the expression further by combining like terms. Like terms are terms in an expression that have the same variable part, with only the coefficients and constants differing.

For example, in our expression, after distribution, we have \(2x - 10 + 15 - 3x\).
The like terms here are:

• \(2x\) and \(-3x\) (both terms contain the variable \(x\))
• \(-10\) and \(15\) (both are constants)

Combine these terms by adding their coefficients:

• The \(x\) terms: \(2x - 3x = -x\)
• The constant terms: \(-10 + 15 = 5\)

By combining like terms, you reduce the number of separate parts in an expression, making it more manageable and closer to its simplest form.

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operations, but no equals sign. These expressions are the foundation of algebra and are used to model real-world situations.

The original expression \(2(x-5)+3(5-x)\) is an example of an algebraic expression. It contains:

• Numbers, such as 2, 5, and 3, which are constants.
• Variables, like \(x\), which are symbols used to represent unknown values.
• Operations, in this case, multiplication and addition.

Working with algebraic expressions involves simplifying them by using properties like distribution and combining like terms. This allows us to convert a complex expression into something simpler, like \(-x + 5\).

Understanding algebraic expressions and their components helps in problem-solving, enabling you to interpret and manipulate mathematical information efficiently.