Problem 52 - 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. $$ -(q-2+6 r) $$

Step-by-Step Solution

Answer

The expression simplifies to \[-q + 2 - 6r\].

Step 1: Apply the Distributive Property

The distributive property states that you multiply each term inside the parentheses by the factor outside. Here, the factor is \[-1 \times (q - 2 + 6r)\]This means multiplying \[-1\] by each term inside the parentheses: \(q, -2,\) and \(6r\).

Step 2: Distribute the Factor

Multiply \(-1\) by each term inside the parentheses:\(-1 \times q = -q\)\(-1 \times (-2) = 2\)\(-1 \times 6r = -6r\)The expression becomes:\[-q + 2 - 6r\]

Step 3: Simplify the Expression

Now, combine like terms if possible to simplify the expression. Since no terms are alike, \(-q + 2 - 6r\)is already simplified.

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions is an essential skill when working with mathematics. It involves breaking down expressions into their simplest form. The main purpose of this simplification is to make the expressions easier to handle and understand. When simplifying, the goal is to combine all the necessary steps to reduce the expression to its easiest version while maintaining the original value. One of the most fundamental tools for simplifying expressions is the Distributive Property. This property allows you to remove parentheses in expressions by distributing a multiplier to each term inside the parentheses. Once the parentheses are removed, you can proceed with any further operations, such as combining like terms if applicable. Understanding this process is key because simplifying expressions is crucial in solving equations, factoring, and even calculus. This is a building block for more advanced algebra topics. Don't worry if it feels a bit complicated at first. With practice, simplifying expressions will become a smoother, more intuitive process.

Combining Like Terms

An important part of simplifying algebraic expressions is combining like terms. You might be asking, what are 'like terms'? These are terms that have the same variable raised to the same power. For example, in the expression \(3x + 5x\), \(3x\) and \(5x\) are like terms because they both contain the variable \(x\). By combining them, the expression simplifies to \(8x\).This technique is very helpful in reducing the complexity of an algebraic expression. Here's how you can approach it:

• Identify like terms in the expression.
• Add or subtract coefficients (the numerical parts) of these terms together.

In the exercise provided, we encountered terms like \(-q\), \(2\), and \(-6r\). Unfortunately, they are not like terms because they don't share the same variable, which means no further simplification through combining is possible.Combining like terms is especially useful in larger expressions where it helps to significantly shrink down the equation into a more manageable size.

Multiplying Negative Numbers

Negative numbers can sometimes be a source of confusion, but multiplying them follows straightforward rules. When multiplying a negative number by a positive number, the result is always negative. However, if you multiply two negative numbers together, the result is positive.In our exercise, we used the distributive property by multiplying each term in the parentheses by \(-1\). Let's break down what happens:

• \(-1 \times q = -q\), resulting in a negative term.
• \(-1 \times (-2) = 2\), where the negative signs cancel out, resulting in a positive term.
• \(-1 \times 6r = -6r\), maintaining the negative because one factor is negative.

These rules are quite useful in algebra and beyond, providing a consistent framework for dealing with negativity in numbers. They help ensure that when you encounter multiple terms in an algebraic expression, you can confidently determine sign changes and simplify accurately.