Problem 51 - Beginning and Intermediate Algebra

Question

Use the distributive property to write each expression without parentheses. Then simplify the result. See Example 4. \(-(r-3-7 p)\)

Step-by-Step Solution

Answer

The expression simplifies to \(-r + 3 + 7p\).

Step 1: Apply the Distributive Property

The distributive property states: \( a(b + c) = ab + ac \). In this exercise, you have the expression \(-(r - 3 - 7p)\).Use the distributive property by multiplying \(-1\) with each term inside the parentheses:\(-1 \times r - (-1 \times 3) - (-1 \times 7p)\).

Step 2: Simplify Each Term

Now, simplify each term that resulted from the distribution:\(-1 \times r = -r\)\(-1 \times (-3) = 3\)\(-1 \times (-7p) = 7p\).

Step 3: Combine Simplified Terms

Write the simplified terms together in a single expression:\(-r + 3 + 7p\). This is the expression with no parentheses, combining all the distributed terms.

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra makes them easier to understand and work with. When we simplify an expression, we're essentially making it as "short" or "clean" as possible. This often involves removing parentheses and combining like terms.
To start simplifying, first deal with any parentheses in the expression. Follow the order of operations, which suggests handling operations inside parentheses first. Once the parentheses are addressed, it's time to deal with multiplication and division, followed by addition and subtraction.

• Simplifying helps in solving equations faster.
• It reduces the number of operations we need.
• Makes comparisons between expressions straightforward.

Remember, the goal of simplifying is to maintain the same value of the expression while making it easier to read and work with. Keeping your expressions simplified is a critical skill, especially when the expressions become more complex.

Distributing and Combining Like Terms

Distributing involves spreading a term outside the parentheses over terms inside the parentheses using multiplication. The distributive property is central to this process: if you have an expression of the form \(a(b + c)\), it should be written as \(ab + ac\).
In the given exercise, apply the distributive property on \(-(r - 3 - 7p)\). Multiply each term by \(-1\) to distribute it correctly:

• \(-1 \times r\),
• \(-1 \times (-3)\),
• \(-1 \times (-7p)\).

After distributing, focus on combining like terms. Like terms are terms that have the same variable raised to the same power. In some expressions, you may have only numbers to combine or variables that multiply by the same power.
Combining these results gives you your simplified expression, helping you keep track of your calculations and clarifying what each part of the expression represents.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations. Unlike an equation, an algebraic expression does not include an equality sign but can still be manipulated using algebra's rules.
Algebraic expressions are the building blocks of algebra. They allow representation of real-world situations mathematically.

• Variables in expressions stand in for unknown values.
• Constants are fixed values within the expression.
• Operations combine variables and constants in many ways.

Understanding algebraic expressions is key to solving more complex algebra problems. By mastering the process of rearranging and simplifying these expressions, you become better prepared to solve equations and understand their real-life applications.
Being comfortable with terms like variables, constants, and coefficients, as well as understanding how operations affect them, sets a strong foundation in learning algebra effectively. So, a deep dive into algebraic expressions will empower you to tackle problems with greater confidence.