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Chapter 1: Problem 313

In the following exercises, simplify. $$ 43 m+(-12 n)+(-16 m)+(-9 n) $$

Short Answer

Expert verified
27m - 21n

Step by step solution

01

Identify Like Terms

Group the terms involving the same variable together. In this case, group the terms with 'm' and the terms with 'n'.
02

Combine Like Terms

Add the coefficients of the like terms together: For 'm': 43m + (-16m) Combine: 43m - 16m = 27m For 'n': -12n + (-9n) Combine: -12n - 9n = -21n
03

Write the Simplified Expression

Put the simplified terms together to get the final simplified expression: 27m - 21n

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Key Concepts

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

Combining Like Terms
Simplifying algebraic expressions often involves 'combining like terms'. Like terms are terms that have the same variable raised to the same power. In the expression given, we had terms with 'm' and terms with 'n'. Both groupings are composed of the same variable, which allows us to combine them. This makes expressions easier to work with, as they become shorter and simpler. To successfully combine like terms, ensure they have the same variable and exponent. This keeps the terms mathematically equivalent while simplifying the expression.
Coefficients
Understanding coefficients is crucial in algebra. A coefficient is a numerical or constant quantity placed before and multiplying the variable in an algebraic expression. In the expression \(43m + (-12n) + (-16m) + (-9n)\), '43' and '-16' are the coefficients of 'm', while '-12' and '-9' are the coefficients of 'n'. When combining like terms, you only add or subtract the coefficients, not the variables. This keeps the terms standardized and consistent. If we simplify, the coefficients help us to quickly combine terms like so: \(43m - 16m\) becomes \(27m\).
Variable Terms
Variable terms in an expression refer to components that contain variables. In the original expression, \(43m\) and \(-16m\) are terms with the variable 'm'. Similarly, \(-12n\) and \(-9n\) are terms with the variable 'n'. When simplifying expressions, it is important to focus on grouping and combining these variable terms correctly. This process respects the mathematical properties of the variables and their coefficients. By accurately identifying and combining variable terms, we effectively reduce the complexity of algebraic expressions.
Simplification Steps
The process of simplifying an algebraic expression can be broken down into clear, easy-to-follow steps:
  • Step 1: Identify and group like terms.
  • Step 2: Combine the coefficients of like terms. For example, from \(43m \text{ and } -16m \), we obtain \(27m\)
  • Step 3: Write down the simplified expression.
In our provided example, after combining the coefficients, we have \(27m - 21n\). Simplification not only makes the equation look cleaner but also makes further algebraic processes easier.

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