Problem 93 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. $$ -9.8 c+6.2 c $$

Step-by-Step Solution

Answer

The simplified expression is \\(-3.6c\\).

Step 1: Identify Like Terms

The expression \(-9.8c + 6.2c\) consists of two terms: \(-9.8c\) and \(+6.2c\). Both terms have the same variable part, which is \('c'\). This means they are like terms and can be combined by adding their coefficients together.

Step 2: Combine Coefficients

Add the coefficients of the like terms: \(-9.8 + 6.2\). First, deal with \(-9.8\), which is negative, and \(6.2\), which is positive. Subtract \(6.2\) from \(-9.8\), which results in \(-3.6\).

Step 3: Construct the Simplified Expression

Combine the result from adding the coefficients to the shared variable \(c\). Therefore, \(-3.6c\) is the simplified form of the original expression \(-9.8c + 6.2c\).

Key Concepts

Combining Like Terms

Combining like terms is a fundamental skill in algebra that makes expressions simpler and easier to work with. Like terms are those that have the same variable raised to the same power. To combine them, you only need to work with their coefficients, which are the numbers in front of the variables.
For example, in the expression \(-9.8c + 6.2c\), both terms have the variable \(c\), making them like terms. To simplify, focus on their coefficients: \(-9.8\) and \(6.2\). You combine these by simple arithmetic, in this case, subtraction. This simplifies the expression to \(-3.6c\).
Combining like terms reduces the number of terms and streamlines expressions, making it easier to solve or rearrange equations. This process is crucial for solving equations efficiently and correctly.

Coefficients

Coefficients are the numerical part of a term that is multiplied by the variable. They are fundamental in determining the magnitude and direction of a variable's effect in an expression or equation.
In our example expression, \(-9.8c + 6.2c\), we have two coefficients: \(-9.8\) and \(6.2\). These coefficients can tell us how much each term will contribute when the variable is assigned a value.

• Positive coefficients, like \(6.2\), increase the overall value of an expression.
• Negative coefficients, like \(-9.8\), decrease the value.

Manipulating and combining coefficients helps in simplifying expressions and solving equations. Being comfortable with coefficients allows for easier management of complex algebraic expressions.

Variables

Variables are symbols used to represent numbers and are essential in expressing mathematical relationships and solving equations. Typically, variables are represented by letters such as \(x\), \(y\), or in our case, \(c\). They stand in place of unknown values and allow us to create general formulas or expressions.
In the expression \(-9.8c + 6.2c\), \(c\) is the variable common to both terms. Because both terms share the variable, the coefficients of these terms can be combined. Variables play a crucial role because they enable algebraic manipulation, which is essential for solving equations and understanding mathematical relationships.

• Variables allow for the abstraction of particular values.
• They make it possible to solve equations by replacing variables with specific numbers.

Mastering the use of variables is key to success in algebra and beyond. They provide flexibility and generality in mathematical modeling and problem-solving.