Problem 82 - Intermediate Algebra

Question

Simplify. See Example \(6 .\) $$5(q+7)-3(q-1)-(q+2)$$

Step-by-Step Solution

Answer

The simplified expression is \( q + 36 \).

Step 1: Distribute First Expression

Distribute the 5 across the terms inside the first parenthesis: \[ 5(q + 7) = 5q + 35 \]

Step 2: Distribute Second Expression

Distribute the -3 across the terms inside the second parenthesis: \[ -3(q - 1) = -3q + 3 \]

Step 3: Distribute Third Expression

Distribute the -1 (implicit in front of the parenthesis) across the terms inside the third parenthesis: \[ -(q + 2) = -q - 2 \]

Step 4: Combine All Terms

Combine all terms resulting from each distribution: \[ 5q + 35 - 3q + 3 - q - 2 \]

Step 5: Simplify and Combine Like Terms

Combine like terms:- Combine the q terms: \( (5q - 3q - q) = q \)- Combine the constant terms: \( (35 + 3 - 2 = 36) \)Thus, the simplified expression is:\[ q + 36 \]

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that helps us simplify expressions and solve equations. The main idea is that you can distribute a factor over terms inside parentheses. Let's break it down:

• If you have an expression like \(5(q + 7)\), the number 5 can be distributed to each term inside the parentheses.
• This means you multiply 5 by \(q\) and 5 by 7 separately, resulting in: \(5q + 35\).

The same concept applies to expressions with negative operations as well. For instance, in \(-3(q - 1)\), you distribute \(-3\) to both \(q\) and \(-1\), resulting in \(-3q + 3\).

• A useful tip: when a negative sign is outside the parentheses, like in \(-(q + 2)\), you can think of it as multiplying by \(-1\).
• So, here you distribute \(-1\) to both \(q\) and 2, leading to \(-q - 2\).

This property is incredibly useful for breaking down and simplifying more complex algebraic expressions, making it an indispensable tool in problem-solving.

Like Terms

When simplifying algebraic expressions, combining like terms is a key technique. Like terms are terms that have the exact same variables raised to the same power. For example, in the expression \(5q + 35 - 3q + 3 - q - 2\), the terms \(5q, -3q, \, \text{and} \, -q\) are like terms because they all contain the variable \(q\).

• To combine like terms, simply add or subtract the coefficients (the numbers in front of the variables).
• In our example, you combine \(5q - 3q - q\) to get \(q\).

Constant terms, like 35, 3, and -2 are also considered like terms because they do not contain any variables.

• These can be combined by straightforward addition or subtraction, resulting in 36 in the given example.

By grouping and combining like terms, you make expressions simpler and more manageable to work with.

Simplification of Expressions

Simplification of expressions is an essential skill in algebra, allowing us to express complex problems in more straightforward terms. By using the distributive property and combining like terms, we can transform a lengthy expression into a more concise and understandable format.

• The goal of simplifying is to reduce the expression to its simplest form by performing operations and eliminating unnecessary terms.

Take the expression \(5(q+7)-3(q-1)-(q+2)\). After applying the distributive property and combining like terms, this complex expression simplifies to \(q + 36\).
Visualizing the steps:

• First, distribute each factor across the terms in parentheses.
• Next, combine the like terms by adding or subtracting the coefficients.
• Finally, you reach the simplest form, making it easier to evaluate or use in further calculations.

This process not only makes mathematical operations more orderly but also enhances your problem-solving efficiency. Understanding and mastering the simplification of expressions equip you with the tools to tackle more advanced algebraic concepts in the future.