Problem 66 - Elementary Algebra

Question

\(15 d-(6 d+1)-d\)

Step-by-Step Solution

Answer

The simplified expression is \(8d - 1\).

Step 1: Simplify Inside the Parentheses

First, simplify the expression inside the parentheses. The expression is \(6d + 1\).

Step 2: Distribute the Negative Sign

Next, distribute the negative sign from \(-(6d + 1)\) to each term inside the parentheses: \(-(6d + 1) = -6d - 1\).

Step 3: Rewrite the Expression

Rewrite the original expression by replacing \(-(6d + 1)\) with \(-6d - 1\): \(15d - 6d - 1 - d\).

Step 4: Combine Like Terms

Combine the like terms. This involves adding or subtracting the coefficients of the same variable:\(15d - 6d - d = 8d\). The resulting expression is \(8d - 1\).

Key Concepts

Distributive Property

The distributive property is a valuable algebraic tool, essential for solving various expressions. It allows you to multiply a single term across terms within parentheses.
For instance, in the exercise: \(15d - (6d + 1) - d\), the distributive property helps when we deal with \(-(6d + 1)\). Here's why:

• This means you multiply \(-1\) by each term in the parentheses.
  

So, \(-(6d + 1)\) becomes \(-6d - 1\). This step ensures each term inside the parentheses is accurately subtracted from the main expression.
It’s a crucial process that makes expressions manageable and prepares them for further simplification.

Combining Like Terms

Combining like terms is all about making expressions simpler.
Like terms are terms that have the same variable and exponent. They can be combined by adding or subtracting their coefficients. In the improved expression \(15d - 6d - 1 - d\), we have:

• \(15d - 6d - d\)

These are like terms because they all have the variable \(d\). Now, let's combine them step by step:

• First, add \(15d\) and \(-6d\), which gives \(9d\).
• Then, subtract \(d\) from \(9d\), resulting in \(8d\).

The final simplified expression is \(8d - 1\). Combining like terms efficiently reduces complexity and solves algebraic expressions more directly.

Algebraic Expressions

Algebraic expressions are a fundamental concept in algebra.
They consist of numbers, variables, and operations (such as addition and subtraction). Our exercise deals with this expression: \(15d - (6d + 1) - d\). Breaking down complex expressions is key in algebra:

• First, deal with the parentheses using the distributive property.
  
• Second, rewrite the expression by combining simplified parts.
• Third, combine like terms to further reduce the expression.

Algebraic expressions are the building blocks of many mathematical concepts.
Understanding how to manipulate and simplify them is essential for tackling more advanced problems later on.