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Chapter 3: Problem 91

Clear parentheses and combine like terms. $$6-(3 m-n)-2(m+8)+5 n$$

Short Answer

Expert verified
-10 - 5m + 6n

Step by step solution

01

Distribute the negative signs into the parentheses

Distribute any negative signs through the expressions in the parentheses: \[6 - (3m - n) - 2(m + 8) + 5n\] becomes \[6 - 3m + n - 2m - 2(8) + 5n\].
02

Simplify by removing the parentheses

Removing the parentheses gives us: \[6 - 3m + n - 2m - 16 + 5n\].
03

Combine like terms

Combine the like terms: constants, terms with \(m\), and terms with \(n\): Constants: \[6 - 16 = -10\] Terms with \(m\): \[-3m - 2m = -5m\] Terms with \(n\): \[n + 5n = 6n\]. So the expression simplifies to: \[-10 - 5m + 6n\].

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Key Concepts

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

prealgebra
Prealgebra lays the foundation for algebra. It's all about basic arithmetic and the rules for manipulating equations and expressions. In prealgebra, you get familiar with operations involving numbers and variables, which are symbols that stand in for numbers. For example, in the expression given, (6 - (3m - n) - 2(m + 8) + 5n), variables such as 'm' and 'n' play a key role. Knowing how to manipulate these expressions correctly is crucial for success in algebra and beyond.
distributive property
The distributive property is a handy tool in algebra that allows you to simplify expressions. It means you distribute a multiplier across terms inside parentheses. For example, in the expression (6 - (3m - n) - 2(m + 8) + 5n), we use the distributive property to handle both the negative signs and the multiplication. We go from 6 - (3m - n) - 2(m + 8) + 5n; to 6 - 3m + n - 2m - 2(8) + 5n. By doing so, each term inside the parentheses is correctly affected by the factors outside.
simplifying expressions
Simplifying expressions means making them easier to work with without changing their value. This often involves removing parentheses and combining like terms. For example, after distributing the negative signs in our given problem, we get: 6 - 3m + n - 2m - 16 + 5n. The next step to simplify this expression is combining like terms to get a cleaner, more manageable result. Ultimately, it simplifies to -10 - 5m + 6n.
constants
Constants are numbers without variables. They're fixed values. In the provided expression 6 - 3m + n - 2m - 16 + 5n, the constants are 6 and -16. To combine constants, you simply perform the arithmetic operations. Here, 6 - 16 = -10. It鈥檚 essential to combine constants correctly to simplify your expressions accurately.
terms with variables
Terms with variables are parts of expressions that include variables along with their coefficients. In our problem, terms with variables include -3m, n, -2m, and 5n. When simplifying, you combine these terms by adding or subtracting coefficients with the same variables. So, -3m - 2m equals -5m, and n + 5n equals 6n. Combining like terms helps in simplifying complex expressions efficiently.

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