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Chapter 1: Problem 77

Combine like terms. \(6 m+9 n+(-9 n)+(-10 m)\)

Short Answer

Expert verified
-4m

Step by step solution

01

Identify like terms

In the expression \(6m + 9n + (-9n) + (-10m)\), the like terms are the terms with the same variable. Here, \(6m\) and \(-10m\) are like terms, and \(9n\) and \(-9n\) are like terms.
02

Combine like terms involving m

Combine the coefficients of the like terms involving \(m\): \(6m + (-10m) = (6 - 10)m = -4m\).
03

Combine like terms involving n

Combine the coefficients of the like terms involving \(n\): \(9n + (-9n) = (9 - 9)n = 0n = 0\).
04

Write the simplified expression

After combining the like terms, we get the simplified expression: \(-4m + 0\). Since adding zero does not change the value, the final simplified expression is \(-4m\).

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Key Concepts

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

Identifying Like Terms
Identifying like terms is a crucial step in simplifying algebraic expressions. Like terms are terms that have the exact same variable part. This means they have the same variables raised to the same power. For example, in the expression \(6m + 9n + (-9n) + (-10m)\), the terms \(6m\) and \(-10m\) are like terms, because they both contain the variable \(m\). Similarly, \(9n\) and \(-9n\) are like terms because they both contain the variable \(n\). Identifying like terms allows us to combine them to further simplify the expression.
Coefficients
The coefficient of a term is the numerical part of the term that is multiplied by the variable. For example, in the term \(6m\), the number 6 is the coefficient, and in the term \(-10m\), the number -10 is the coefficient. When combining like terms, you primarily work with the coefficients. Here's how we handle them in our example:
  • For the terms involving \(m\): \(6m\) and \(-10m\), we combine the coefficients 6 and -10, which yields \((6 - 10)m = -4m\).
  • For the terms involving \(n\): \(9n\) and \(-9n\), we combine the coefficients 9 and -9, giving us \((9 - 9)n = 0\).
After this step, we notice that the term with the coefficient 0 can be removed because multiplying anything by 0 results in 0.
Simplifying Expressions
Simplifying expressions involves combining like terms and performing basic arithmetic to arrive at a more condensed form of the expression. Let's look at our original expression: \(6m + 9n + (-9n) + (-10m)\). After identifying and combining the like terms (as explained in previous sections), we had:
  • \(6m + (-10m) = -4m\)
  • \(9n + (-9n) = 0n = 0\)
We then write the simplified expression, which is \(-4m + 0\). Since adding 0 does not change the value, we can further simplify this to \(-4m\). The simplified expression effectively represents the result after combining all like terms and eliminating any terms that sum to zero.

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