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Chapter 5: Problem 60

Add or subtract as indicated. $$ (2 d+7)-(3 d-1) $$

Short Answer

Expert verified
-d + 8

Step by step solution

01

Distribute the subtraction

Apply the distributive property to remove the parentheses. This means to distribute the negative sign through the second polynomial: \((2d + 7) - (3d - 1)\) becomes \((2d + 7) - 3d + 1\).
02

Combine like terms

Identify and combine the like terms. Here, the terms with the variable \(d\) are \(2d\) and \(-3d\). The constant terms are \(7\) and \(1\).
03

Perform the addition/subtraction

For the \(d\) terms: \(2d - 3d = -d\). For the constants: \(7 + 1 = 8\).
04

Write the simplified expression

Combine the results of the previous step to get the final simplified expression: \(-d + 8\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a key concept in algebra. It allows us to multiply a single term across multiple terms within parentheses. This property helps in simplifying expressions and solving equations. In the given exercise, we applied it to the subtraction of polynomials.
For example, we start with the expression \( (2d + 7) - (3d - 1) \). We need to distribute the negative sign across the second polynomial. This turns the expression into \( (2d + 7) - 3d + 1 \).
This step is crucial as it sets up the next phase鈥攃ombining like terms.
Combining Like Terms
Combining like terms is another essential skill in algebra. It involves merging terms that have the same variables raised to the same power. In the exercise, we identified and combined the terms with the variable \( d \) and the constant terms separately.
Our expression after distribution was \( 2d + 7 - 3d + 1 \). We look at the \( d \) terms: \( 2d \) and \(-3d \). Combining these, we get \( 2d - 3d = -d \).
Next, we add the constant terms \( 7 \) and \( 1 \), giving us \ 7 + 1 = 8 \.
Simplifying Expressions
Simplifying expressions means reducing an expression to its simplest form. After combining like terms, we put together our results.
For the given problem, we had \( -d \) from combining our \( d \) terms, and \( 8 \) from adding the constants. Putting it all together, we get the simplified expression: \ -d + 8 \.
This final expression is easier to understand and use in further calculations. Simplifying expressions helps to make complex problems more manageable and is a fundamental process in algebra.

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