Problem 75 Simplify each expression. $$ ... [FREE SOLUTION]

Chapter 1: Problem 75

Simplify each expression. $$ -5 b-(-a t+7 b) $$

Short Answer

Expert verified

The simplified expression is at - 12b.

Step by step solution

Distribute the Negative Sign

Rewrite the expression by distributing the negative sign inside the parentheses. The expression becomes: -5b - (-at + 7b) which simplifies to: -5b + at - 7b

Combine Like Terms

Identify and combine the like terms. In this case, the like terms are -5b and -7b. Combine these to get: -12b and then you have to add the term at. The simplified expression is: at - 12b.

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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 useful tool in algebra that allows you to multiply a single term outside a set of parentheses by each term inside the parentheses. For example, if you see something like $a(b + c)$, you would distribute $a$ to both $b$ and $c$, resulting in $ab + ac$.
In our exercise, we have a negative sign before a set of parentheses: $-(-at + 7b)$. It's important to understand that distributing the negative sign means multiplying each term inside the parentheses by $-1$.
So, $-(-at + 7b)$ becomes $+at - 7b$. This step is crucial as it helps transform complicated expressions into simpler ones, making it easier to combine like terms later.

Combining Like Terms

Combining like terms is another critical skill in algebra. Like terms are terms that have the same variable raised to the same power. For instance, $3x$ and $5x$ are like terms because they both have the variable $x$.
In our exercise, we first distributed the negative sign and ended up with $-5b + at - 7b$. Now, we need to combine the like terms $-5b$ and $-7b$.

$-5b$ and $-7b$ are like terms because they both contain the variable $b$.
When combined, these terms give: $-5b - 7b = -12b$.

This simplifies our expression to: $at - 12b$. Combining like terms helps in reducing the expression to its simplest form, making it more manageable and easy to understand.

Negative Sign Distribution

When you see a negative sign before a term or an expression in parentheses, you need to distribute that negative sign to each term within the parentheses. This process is sometimes called 'negative distribution' or 'distributing the negative sign'.
Think of distributing a negative sign as multiplying each term inside the parentheses by $-1$. For example:

The expression $-(-at + 7b)$ means you multiply $-1$ by each term within the parentheses.
So, $-(-at) = +at$ and $-(+7b) = -7b$.
Thus, $-(-at + 7b)$ becomes $+at - 7b$.

This step is very important because missing out on distributing the negative sign correctly can lead to incorrect results. Make sure to check your work carefully to ensure that all negative signs are properly distributed.

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91影视

Simplify each expression. $$ -5 b-(-a t+7 b) $$

Short Answer

Step by step solution

Distribute the Negative Sign

Combine Like Terms

Key Concepts

Distributive Property

Combining Like Terms

Negative Sign Distribution

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